You will find Lindau's harbour on the island. If you want to go for a boat trip or take some unforgettable pictures, please go there, even though the traffic is heavy. You won't regret it. You can use public transportation if you are afraid not of not finding a parking spot. Most of the attractions are there.
There are two light towers - one of them built in the 13th. century - and many other interesting attractions. Climb up the newer light tower and get a great view - weather permitting of course.
Go into the "Alte Rathaus" (old city hall) which dates back to the 15th. century.
Just browse around and absorb the atmosphere.
It is your decision whether you want to try your luck in the casino (Bayerische Spielbank). Be prepared to face a dress code.

Lindau satisfies all your accommodation needs. You must not rush to see all the attractions and to enjoy fine dining. You can go to museums, listen to concerts or be more active by partaking in available sports.
Many other places can easily be reached. You are very close to Austria. Have you ever been in Bregenz?
It is also not too far to Switzerland and Liechtenstein. Or explore the German part of Lake Constance. Just ask for information.