How to use public transport in Russia with limited knowledge of Russian?

I spent 2 weeks in Russia around 2 years ago, so here is my advice.

I learned to (sort of) read the Cyrillic alphabet (helped me a lot and, believe, it's not that difficult), but what really saved me during the rides was Google Translate app.

Before taking any trip, I would access internet at the hotel and pre-plan my ride. Like "I gotta use Metro's green line heading ?????????? and leave at ??????????? station". In the Metro being able to read Cyrillic and paying attention to every stop was enough to get me where I wanted.

When I was in buses (?????????) I would use Google Translate app and write something like "Excuse me, could you please tell me when we arrive at the closest stop to PeterHof Castle?". The app translates that to Russian and if you tilt your cell phone to horizontal, the sentence would be shown in full-screen. Every time I did that Russian from all ages and genders would gladly help me, trying to communicate and making sure I left the bus at the correct stop.

In short, Russians where very helpful, but I hardly found someone who could speak English (not even a policeman with a big tourist police on his uniform, near the Kremlin did). Technology solved most my communication issues.

Just to add to already present excellent answers:

The most convenient mode of transport in Russia should be ride hailing. Just call Uber or Yandex Taxi from your location to your destination. Usually it works flawlessly. This includes driving from/to the airports.

For everything else, you should find it easier to build public transportation routes using Yandex Maps or 2GIS. Even if they are in Russian, most of stuff on the screen is icons anyway. In Russia, you can usually buy tickets from the driver or conductor in buses, trams, trolley busses and route taxis. The only exception here is railway, where you should be getting your ticket before boarding the train.

There are very good answers already; here is my small contribution. I know literally no Russian at all. And that didn't cause me any problems whatsoever while spending time in St. Petersburg and Moscow, and travelling in between.

As mentioned, planning helps a lot; and it also helped that I had an app (Yandex?) I found that had the whole metro map with station names both in Cyrillic and Latin characters.

If I had stayed a bit longer I would have bought a sim and data for my phone, to be able to do searches and translate; but over a few days I didn't really feel the need.

Never hitchhike in Russia. It can be very, very dangerous. Never use unlicensed taxis.
Using a licensed taxi, the best option is to have the address written on the paper. The driver will just put it into his navigator.

Unless you speak fluent Russian (or unless you are traveling to the end of the route), don't use marshrutka (minibuses). This is a very convenient transportation, but you need to communicate with the driver. It is important that you tell him the exact point you need. Sometimes you need also to keep your eye on the route. Remember that the driver sometimes has to keep in mind several destinations from the passengers. So drivers make mistakes regularly. In fact, this is the best possible kind of transportation for traveling the same route every day. But the first use of a route can be challenging.

I also recommend you to have a map near you. Even if you don't find an English-speaking passer-by, you will always find somebody who'll be able to give you the right direction on your map

AND remember Crime and violence in Russia are not as great as some western people think. What is really dangerous here is to be a pedestrian. Our drivers are often careless

In my experience very few Russians know English, so it's a problem.

I can only suggest learning some Russian or translating software.

RULE #1: don't be too trusting, or think women are hitting on you. Russia has a huge crime problem. If they know you're a foreigner, you'll be a target.

You can always install the Google Translate app on your phone. It does real-time visual translation of signs in a variety of languages, so all you need to do is hold your phone's camera up to a sign to see its English translation.

In the event you need to talk to a conductor or ticket agent, you can also use it to translate speech and/or written messages.

It may not be the best general solution for communication in a language you don't understand, but for a limited task like taking public transport, it should be more than sufficient.