Lindau is just full of scenic spots, particularly the harbor with the Mangturm and the harbor entrance with the lighthouse and lion (the symbol of Bavaria) statue. You can get a good picture from the Hafenplatz looking across the harbor, through the entrance, past the lighthouse, at the mountains of Switzerland.
Also there's the view down almost any street in the interior, the Diebsturm (thieves' tower, a prison), the old city hall in Bismarkplatz.
As for other things to do, we took a boat across Bodensee to Rorschach, Switzerland, then came back to Lindau around the east end of the lake, stopping in Bregenz, Austria, to go up the Pfänderbahn from where we could see all the way down the lake to Konstanz.
We changed trains in Tübingen on our way from Sigmaringen to Herrenberg. We walked a short distance to the shore of the Neckar and had lunch at a riverside restaurant, sitting on a deck overlooking the river watching the "punts".