There's a nice little town near Lindau, name escapes me, Wasserburg perhaps? Worth checking out too.
Meersburg. Very beautiful and a priority.
Lindau is also very attractive. A suggested itinerary would be: A look around Lindau on the first morning, a cycle ride along the lake in the afternoon. You can get the train back once you've had enough. On the second day, visit
Meersburg, perhaps detouring to
Kloster Birnau on the way. Or take a ferry ride across to Konstanz and have a look-see around the town.
Any spare time can be spent eating ice cream etc; it's a nice area to unwind. Personally, I wouldn't bother about Bregenz.

Meersburg