Traveling from Germany to other countries by train?

You should always have your passport with you when out of your own country.  Within Schengen, you’ll not likely be asked for it, but it’s possible, and if you don’t have it when asked, definitely a hassle or worse.

You can probably book train trips on whatever site you use within Germany, but you can also easily do it at Loco.

There are probably other sites.  Rick Steves has some good informational pages about trains (and other travel info) in Europe.

The train between Munich and Innsbruck through Kufstein has spot checks. It doesn't stop but the border police enters the train a few stations before Kufstein and walks through the carriages.

If you look suspicious, they want to check your identity so have your passport at hand. There are no customs checks.

It is extremely easy and there is no check.

As a non-Schengen national I took the train between Austria and Germany twice in the last few months, and there was no suggestion that passports might be checked. I also took a bus tour that crossed the border and we were not even advised to bring passports. There are not normally checks when crossing the border between Schengen countries - that's the point of the Schengen area.

That's possible, and very easy.

so is it basically just like taking the train within Germany then? Do I even need to bring my passport?

Sadly, it is not just like taking a train within Germany. In theory, it should be. In practice, due to developments since 2015 onwards, no. You may be, and should expect to be controlled at the border. And you had better had a passport on you then.

Regardless, you should carry your passport anyway. While at least in Germany there is no general obligation to carry ID at all times (or passport in your case, since you are not a Union member, so you have "Passpflicht", not "Ausweispflicht" -- which means ID is not sufficient), you are required to be able to prove proper ownership of a passport and you are to present it upon being asked by police in due time. Well, whatever that means... in due time (it's not defined, I am assuming something like "Sure, let me go inside my room and fetch it" would do, but "Uh, I left it in another city" might not). Not having your passport on you doesn't mean you're going to get sentenced to death, but well, it means trouble which is not really necessary during your holiday.

Plus, when crossing borders, carrying your passport is a strict requirement.

somewhere in Austria (Innsbruck maybe?) so I could see the Alps.

If the intent is to literally see the Alps, you can do that just fine even without leaving Munich. Or, you can do it from a ton of very lovely nearby places all over the region which are easier to reach (Tegernsee or Chiemgau come to mind, or even some nearby places which you can actually reach by subway or suburban train, Munich area is quite awesome in that respect).

I'm not saying you shouldn't go to Innsbruck, it's definitively a good choice. Though depending on your luck, and depending on how late you buy, you may be surprised how much Deutsche Bahn can charge for a ticket.

Also, there's actual Alps in Germany (think e.g. Zugspitze), so you do not absolutely need to cross a border to see (in a sense of walking them) the Alps. It's somewhat closer (half an hour train time less), too.

If Kufstein (Austria) would also do instead of Innsbruck, you can possibly save quite a bit of money by getting a "Bayernticket". That's a very interesting option in any case because it's a 24-hour do-what-you-want ticket (no ICE, all within all of Bavaria, plus almost -- but not quite -- to the Lake Constance (2km short, that's probably on purpose!), plus some locations in Austria -- Kufstein and Salzburg among them, but not Innsbruck) for a very reasonable price, and if you travel with family or friends, you can add them to the ticket for a small extra charge, which makes it very worthwhile.

Travelling regularly between Germany and Italy I usually get stopped at custom when entering Switzerland, a Schengen Area country but not part of the EU, but never happened when crossing Austria. There should not be any barrier except due diligence police controls on trains and at boarding in the train station.

Edit: corrected as Switzerland is actually part of Schengen Area but not part of EU Schengen info

If your intention is to cross the border from Germany to Austria, that is easy once you are legally inside the Schengen area (includes all of the continental EU except Croatia, Romania and Bulgaria, plus non EU members such as Switzerland and Norway). Systematic border checks have been abolished and the entire EU is a single customs union so there should be no ID checks and there are never customs checks regardless of whether you are on an aeroplane, on a train, in a car, on a bicycle or on foot. With the refugee arrivals of 2015, ID checks have been temporarily reintroduced at some places such as Freilassing but even if you are subject to inspection it should be a rapid one. You should, border checks or not, always have your passport with you when you leave your hotel/accomodation as both Germany and Austria have some form of ID requirement: by law, you don’t necessarily have to carry it with you at all times (at least in Germany) but you should be able to produce it rapidly upon request.

I can attest that Innsbruck is a beautiful city and should be visited. However, if your goal is really just the Alps then:

• they can be seen from Munich in the correct weather conditions (Föhn wind)
• they can be seen on any sufficiently clear day from somewhere just south of Munich such as Seeshaupt on Lake Starnberg, Weilheim i. OB. (denoted as Weilheim(Oberbay) by DB) or Murnau. Aside from the Alps there is also beautiful nature in that area.
• they extend into Germany so you can get the full mountain experience by remaining in Germany. I suggest visiting Garmisch-Partenkirchen which in itself is also a beautiful town (not legally a town though) and features the ski jump arena used for the Vierschanzentournee, the Partnachklamm gorge and various ropeways/one rack railway up onto the various surrounding mountains. Garmisch-Partenkirchen is also at the foot of the Zugspitze, Germany’s highest mountain (although the mountains you can see in Austria are higher).

Concerning your rail trip, if you’re going to Innsbruck directly, the fastest route is (sadly) the one via Rosenheim and Kufstein. This route goes through the flat land north of the Alps to Rosenheim and then through the wide Inn valley into Innsbruck. However, the train passing Garmisch-Partenkirchen also connects to Innsbruck via the Karwendelbahn. It may not be as fast but it offers wonderful scenery on your descent from Hochzirl down into Innsbruck central station. Furthermore, this trip leaves the flat country sooner and goes through smaller valleys via Farchant and Mittenwald. A discounted ticket exists to go from Munich via Garmisch-Partenkirchen to Innsbruck (but I am too out of the loop right now to give you the definite cheapest option).

Disclaimer and source: My family lives between Munich and Garmisch-Partenkirchen.

Adding to what others have said, theoretically there are no systematic border checks between Schengen countries. However, there are frequent "random" checks. I live in Germany and have travelled 4 or 5 times in the last year to Poland and Czech Republic via train and have been checked for a passport twice.

With respect to Austria specifically, there are temporary border controls in place between Austria and Germany until November of this year. There's some diplomatic language in there about under what circumstances they will be in force, but suffice it to say, its not unlikely that you could be checked.