11 Days Around Munich in May

A standard source for moving from zero ideas to some reasonable choices is the one below, provided by Rick Steves himself. Rick focuses too much on Bavaria, IMHO, but since Munich is actually in Bavaria, some of his suggestions may suit you. I would also consult a few guidebooks from other authors,

https://www.ricksteves.com/europe/germany/itinerary

Crossing the borders any real distance would bring up the subject of highway toll stickers in Switzerland and Austria, which are not a deal breaker and can be worked around - but frankly you don't need the hassle of learning the systems because Bavaria is big enough. They have themed roads, which in RS parlance are self guided tours by car. Have a look at https://www.romanticroadgermany.com/ and https://www.deutsche-alpenstrasse.de/en/home . You could follow the first to Füssen and then the eastern part of the second to almost Salzburg.

Starting in Munich we’ve done this… Berchtesgaden area/Salzburg to Hall In Tirol/Innsbruck area to Mittenwald back to Munich.

We were recently in Munich for an extended stay and took day trip to Salzburg that was amazing. Exceeded expectations. We also spent three days in Garmisch Partenkirchen a short 1.5 hr away from Munich and did cable car ride up the Zugspitze mountain the highest peak in Germany, and did short hike around the turquoise Eibsee lake at its base. We also toured Linderhof Palace on our drive to Garmisch. It was beautiful small but exquisite manicured grounds. It was King Ludwig's summer residence. I wish we had more time to spend in Garmisch. I would have taken a short trip to Mittenwald the violin makers dream.

Lake Chiemsee ca. 1h from Munich by car or train

Boatrip to the Islands Herreninsel(Palace) and Fraueninsel(Monastery)

or

Mittenwald 1,5hrs from Munich by car via Benediktbeuren Monastery Lake Kochelsee and Lake Walchensee