Is it a good idea to go to Japan without a travel guide?

You can. Travel guides can be helpful in many areas but Japan is so safe that even doing the wrong thing will not get you in trouble. Getting lost might happen from time to time but it's part of the fun. Japan is known for having strict etiquette but they are very forgiving of foreigners and can get the gist by observing others before doing something.

What I highly recommend is getting a good guide book. It will be a great help for many reasons:

• Planning ahead is crucial to get the best use of your time and the best value which is important if you are on a tight budget. It is easy to spend significantly more when you arrive on-site if you are not aware of prices, you are not going lower-cost seasons and you are not taking advantage of tourist programs that are there to lower cost (such as bundles giving access to multiple sites, transport,etc). A good guide will cover all of those, just be sure that it is no more than a few years since being published.
• Having the guide with you, even in ebook form, is really helpful. When asking for directions in places that use non-Latin letters in their common writing, this is extremely important. You can show a page when you ask for directions and locals will be much more able to help since most guides will show the anglicized name besides the name in the local language.
• Guides offer images, maps, sometimes extensive paper maps and usually include transit maps for place that have massive transit systems.

The web can provide much of that information for free but it will be scattered and of varying levels of accuracy, as most things online are! A guidebook costs very little but gathers everything in one place, including tips and common phrases. If you have a smartphone, you can bring it too but support could be troublesome since bands supported by your phone may not work in Japan, particularly if you are using a CDMA carrier, so download everything possible in advance (maps, addresses of points-of-interest, tickets, etc).

My personal favorite guides are from Fodor and they have served me better than any other guides. They have a good amount of information covering typical tourist interest and their organization stands out which makes them easier to follow and refer to than most books. Ebook versions are available too if you do not want a hard copy due its weight.

If by “travel guide” you mean hiring someone to take you to places: this only useful or necessary in areas where it's hard to get around or where there are places you must avoid, and it's only affordable for budget travelers in places where the local currency is weak. Japan has excellent infrastructure and is extremely safe: you don't need a guide. And you can't afford one.

If by “travel guide” you mean a guide book: absolutely, yes. A book (or equivalent, such as Wikivoyage) will help you figure out what places you'd like to visit and how to get there. Time is money when you're traveling, so plan ahead to make best use of your time.

Navigating a place where you can't read is definitely a challenge, but smartphones make it a lot easier. In most of Japan, train stations have signs with the station name in Latin script and departure boards with destinations in Latin script. Most places are reachable by train, so macro navigation is not so hard. Micro navigation is another matter. Most Japanese towns don't even have street names, only block names. Finding the right house requires reading a block map, which is likely to be in Japanese only. Having a smartphone with GPS that can give you directions helps a lot.

Make sure to download offline maps of the areas you'll be visiting, preferably with both Japanese names and Latin transcriptions. Latin transcriptions are usually enough to pronounce the names in a way that locals have a chance of understanding. You can download offline maps from OpenStreetMap in various smartphone apps.

@Itai's answer is great, but I just want to note that it does depend on where you go. Japan is not a country where many people can speak English; not that people don't like foreign tourists (surely there's quite a bit of xenophobic sentiment in Japan, but that's not what I'm talking about here), but many people just can't express themselves in English at all, an unfortunate consequence of the failure of English language education in the country.

If you just want to go to Tokyo, Osaka, Kyoto, etc., and all the touristy places, then you'll do fine. You can count on at least one or two people who can speak basic English.

But if you want to go to the countryside, the "off the beaten path" destinations, I'm not sure if you can count on that anymore. While there are some solutions (e.g. Google Translate), you will run into a language barrier. So, when Japan reopens for tourism, I guess you'd want to try the most touristy destinations first. If you decide that you do like the country, you can return and explore the rest of the country, hopefully after having learned a bit of the language (just a bit will suffice).

P.S. Nowadays Wikivoyage is a great alternative for travel guides. It is free and easily accessible, but it still lacks in terms of details and depth. However, for me, it has worked pretty well, at least for Japan.

P.P.S. I have never heard of anyone having trouble with their phones when coming to Japan in the 4G age. Unlike 3G, for which competing standards were a huge pain when people moved across borders, there is virtually only one standard for 4G (LTE), so the vast majority of phones should be usable. Most devices can operate in most LTE bands, so that would hardly be a problem. Voice may not work if you don't have VoLTE, but (1) most carriers support it nowadays and (2) there's little chance you'd need to do traditional voice calls. 3G network will be shut down in Japan soon, so you'd better have 4G/LTE service!11

I don't think you have to worry too much as an English speaker with a smartphone, a data plan (not too expensive) and Google Translate.

Even without a smartphone translator, plastic food in the window makes it easy to get decent meals (just point). People are generally kind and not likely to lead you astray, sometimes almost helpful to a fault.

One thing to be aware of if you do go on your own (I found out the hard way): if you plan on visiting any museums in Japan like the Studio Ghibli museum or the Epson museum, you have to buy your tickets way in advance, just like if you were buying tickets to a play or a concert. If you plan to visit any of the castles (highly recommended!) make sure you know what hours they're open on which days. Also, look into the Japan Rail Pass: it's a special deal for foreign visitors only that lets you ride unlimited for a week or so. You'll save a lot of money over buying individual tickets.

You can definitely travel in Japan without a guide.
Your mileage will depend a lot on where you go. If you stick to the standard tourist track of Tokyo-Kyoto-Oska-Hiroshima, you'll get around easily with just English.
Anywhere else, you should be more adventurous, and I strongly recommend learning a few sentences in Japanese, and be comfortable communicating with signs etc., and getting lost a bit more. As others have said, it is very safe and getting lost will just be part of the adventure, nothing you need to actually worry about.

I would only consider a guide if you want to go to places that are both off the beaten path, and in a region that is less accessible by public transport, as much of the North-Western coast is. For example, you can eaisly get to Tottori city by train, but a lot of the more interesting sights around Tottori require a car.

Also, in Japan more than other places (in my experience), advanced planning can really reduce your budget a lot. Do a bit more homework before going, it will pay dividends. I do not think a guide book is necessary, simply because there happen to be very well maintained websites for travel in Japan.

Perhaps my first trip there can help you with some ideas/inspiration: https://dokodokobot.travellerspoint.com/1/

"We have never gone anywhere to a foreign country, and the budget is kinda tight"

Puerto Rico to Japan is quite a big first step for a first-time traveler with a tight budget.

There's Spanish-speaking and English-speaking countries that are much closer (for example in the Caribbean islands) and would be much easier for you on your "tight budget", much closer time-zone wise, and much closer culturally.

However, if it's already been decided that the first country you ever visit will be Japan, I don't think hiring a guide is a good use of your tight budget.

Japan Guide and WikiTravel are extremely thorough for Japan travel, and hyperdia is extremely accurate for navigating which trains to take.

I will reiterate though, that personally I would recommend first visiting some closer countries that are much cheaper and where they speak the same language as you, since this traveling experience will be very valuable for your future trip to Japan. I've been to 50+ countries and also lived in Japan from 2013-2016, and Japan was certainly one of the more difficult places to travel, much because of the language barrier but also because of other things that I probably won't be able to explain here, but I can at least list some things:

• This may be surprising to those who haven't been to Japan, but access to WiFi isn't as easy there compared to a lot of other places, and this can make life as a traveler difficult (the WiFi situation in Japan steadily improved while I lived there, but I'd say it was still significantly worse than every country I'd visited other than Cuba, and in Cuba the WiFi might have actually been better than in Japan, it was just more expensive).
• You won't be able to rely on websites the way you're used to. Not only are a lot of them only in Japanese, but even if you had a Japanese speaker with you, a lot of the places you may want to visit won't have online-booking, or information available on websites the way you might be used to. One example that came up for my cousin that was visiting Japan while I lived there, was that she wanted to watch a sumo wrestling match and I needed to help get the tickets, and this leads me to the next point:
• A lot of tickets are purchased through a machine at a convenience store, for which there can be a bit of a learning curve (even the native Japanese employees at the store would often not be able to help me with using this machine).
• Japan also still has very much of a "cash-based" economy, so a lot of places don't accept credit card. To get cash, there's quite a big difference in cost for currency conversion at the typical places that would be easy for tourists (airports, banks, etc.) versus the more "hidden" vendors such as "Tokai" shops, which might be hard to access for a non-Japanese speaking, first-time traveler.
• Google Maps is quite good in Japan, but there's still places that you won't quite be able to find using Google Maps. The language barrier is one thing, but there was even an incident when I was trying to meet friends at a restaurant in Osaka and several Japanese people failed to help me find the place even though they could perfectly read the address. Likewise, when I first moved to Japan several locals tried to help me find my pre-booked hotel/guest-house, with no success, and by the time I got there they wouldn't allow me in because it was passed the latest check-in time, and no other hotels had space for me since it was a national holiday: a 24-hour convenience store allowed me to stay in the store until the employee's shift ended around 5am, at which point I was taken to a police station that helped find me a "capsule" hotel.

It's for those reasons that I would recommend getting some traveling experience in more familiar/easier places first. By no means do I want you to miss out on the amazing and unique experience of visiting Japan, but to get the most out of it, it helps to have some traveling experience first. Overall though, you do not need a guide, and I've actually never heard of someone hiring a guide to travel in Japan.

Putting the curent global health pandemic to one side, Japan is an incredible country to explore - but really difficult on a tight budget.

You definitely don't need a person to guide you, but some decent guidebooks and tons of research before travelling is essential. (In normal times) Many places where you might want to stay or visit (and the trains that take you to them) get booked up months in advance, so it's really important to plan ahead.

I would also really encourage you to learn to read hiragana and katakana (even if it's just the shapes of the letters). While important notices are often written in English, it's really handy to know the written Japanese for words like 'Karaoke' and 'Restaurant'!!

I was in Japan way back in 1975. Zero Japanese. It was not unusual to not know what we were eating, but so long as you don't have must-not-eat things that's not a big deal. Obviously, my memory of that trip isn't too detailed by now.

It would be vastly easier to do now. Several years ago we had two overnight stopovers and even with my wife being rather skittish about strange situations we fared fine, no problem ordering by pointing.

Now, with Google it would be even easier. The only reason I would even consider assistance in any modern country is for dietary reasons. Guides are only needed in places/situations where doing the wrong thing can be dangerous.

I speak only Spanish and English, yet I had no problems on my own in Turkey, Taiwan, Korea, Portugal, Italy. More than two weeks in each. And no problems for shorter visits to dozens of other countries. I suspect Japan would be no different.

(Actually, the Spanish helped in Italy.)

Added as it was posted as a comment:
Turkey: Learned how to ask directions and order food in Turkish. Otherwise, just looked at maps and walked around. Taiwan, Korea, Portugal: most restaurants had English on their menus. Public transportation, including intercity, was no problem. But I also walked a lot. Italy: Knowing Spanish enabled me to have limited communication with Italian speakers.

If you are going to Japan as your first trip to a foreign country, my guess is you're going to try to do touristy stuff and not try to "have experiences". I'll base my answer on that, and restrict everything else I'm saying to what you can expect to find in large cities. Be aware that if you go to small towns, the following information may not be correct.

It's pretty easy to get around in Japan without Japanese, as long as you speak English and you don't need to talk to anyone. Japanese people speak English pretty poorly overall, so trying to ask for anything complicated may land you in trouble. However, all signage in Japan is bilingual in English, so finding directions around the train station, or finding a washroom, or reading (most) maps should be fairly easy. Many restaurants have English menus as well, at least the larger chains, so finding decent food should not be hard either. A note about ordering food in Japan is that it is quite common for foreigners to point at what they want on a menu, so don't be afraid to just point at what you want instead of trying to pronounce it poorly.

In general, you will want a smartphone with an internet connection much more than you'll want to be able to talk to people. Japanese geography is quite confusing, so knowing how to get from point A to point B is important. You can buy a travel SIM card from the larger electronics store chains and also some convenience stores, which should last most of your trip (don't watch YouTube on it though, or you'll find yourself losing valuable data quickly!). My recommendation is to not buy it at the airport from the SIM vendors there; they tend to be very expensive and you can find one for half the price at a store in the city. Once you have that, Google Maps is your best friend, both for foot travel as well as train travel; Google Maps has really good accuracy with Japanese trains. Don't take taxis in Japan; they are very expensive and not particularly more efficient than the train in terms of travel time if you have Google Maps to guide you when to switch trains and so on.

Most Japanese people are pretty understanding of foreigners who don't know Japanese, and they'll do your best to help you out when you're stuck. That said, be aware that they mostly do not actually speak English (not well, anyway), so don't try to ask them for anything complicated or you'll wind up with a ?????? look and no progress. However, they do mean well, so don't be afraid to ask for help if you need it, the same way as you would in your own country.

As for staying out of dangerous situations, there is no such thing as danger in Japan. The crime rate in Japan is extremely low, and you will be 100% safe at any time of day, anywhere in the country, by yourself or in a group. Don't worry about getting randomly mugged in a dark alley or whatever, that sort of thing simply doesn't happen in Japan. Don't worry about carrying large sums of money on you if you think you may need it for some reason; you won't get robbed, and if you do drop your wallet by accident it's more likely for someone to return it, with the cash still intact, than for someone to make off with it. As a Westerner, this concept is fairly foreign to someone who hasn't been to Japan, but really I mean it, you are safer in Japan than probably anywhere else in the world. Just don't talk to anyone wearing a suit in Kabukicho, that's the main way to get yourself in trouble if you don't know what you're doing.