Suggestions for the most worthwhile day trip out of Munich, please.
There is an awful lot to see in Munich (Marienplatz, Deutsches Museum, Nymphenburg, Pinkotheke, Dachau) without even taking a trip outside Munich.
After that, a lot depends on what you want to see.
The castles in Füssen are not that old, but an icon of Germany. For an older castle, go to Burghausen. Or go to Garmisch-Partenkirchen with a connecting trip to Mittenwald, Linderhof, or Oberamergau. Rothenburg is a bit of a distance for a day trip, but Nördlingen is far less touristy and also has a wall and a church tower you can climb. Nürnberg is close and has many historical Nazi sites. You can make a day trip to Salzburg. Part way to Salzburg is the island palace of Herrenchiemsee.
Also, Regensburg and Chiemsee. Nürnberg. Oberstdorf, if you're there at the right time of year.
If you only have one day and want a quick overview of southern Bavaria, I would suggest:
Rent a car early from the Hauptbahnhof and drive down to the Ettal Abbey. Pop in and see the church and then drive another few minutes to Oberammergau. Walk around and explore the town for an hour or two. See the painted houses, the passion play theater, and do a little shopping. Then get back in your car and drive over to Füssen to pick up your reserved tickets to Neuschwanstein. See the castles and hike up to the Marienbrücke. Drive back towards Munich and swing by the Wieskirche. Drop the car back at the Hauptbahnhof and be downtown by dinner.