The Bayern ticket, which is 25€ I believe and good for up to 5 people, covers all public transport in Bavaria, including Salzburg too. Just go to the Hbf in Munich and stand near the shop by the end of the tracks on the north side of the station, I forgot the number of the tracks, but they are probably in the 20s. You will see a bunch of people standing around waiting, and just start asking if any are going to Salzburg. The Bayern ticket though, does not allow you to travel on the ICE, but you can save a ton of money.

My professors in Munich had said Salzburg can be a daytrip if you take the earliest and latest trains there and back, so I don't know how many nights I would stay. Maybe one night if any? I would try to stay in Munich as long as possible though, and just enjoy the biergartens, bier, food, and culture. I would visit Schloß Nymphenburg as well, very nice, and is just outside the city.

2 hours away is Castle Neuschwanstein, and again is accessable via Bayern Ticket, there is also Chiemsee and Linderhof nearby which are supposed to be beautiful, but I haven't visited yet.

I studied for 5 months in Munich, and currently intern in NRW, if you need any more info, just ask, and there are a couple other knowledgeable people on here too.

If you venture a little further, Prague is 4 hours away (2 nights will do it), and you can get there quite cheap with a Bayern-Bohemian ticket which is 33€, again for 5 people, and takes you to the Czech/German border, and from there it is about 8€ on top of that. Can take a night train there or back, or even to Vienna. That two less nights to pay for, and you can visit another city if you want.