Lake Constance/Bodensee

The town of Uberlingen is directly on the shore of the lake. It is less known and therefore not so many tourists. The town is very pleasant and there is a unique fountain near the shore. There is a dock where you can catch the steamers for lake trips.
There are a few hotels just a few steps from the lake shore. My favorite is the Hotel Ochsen, family-run hotel with an excellent restaurant.
Check with Booking.com for other suggestions.
I have only traveled there by train, but many people come by car.
Hope this is helpful.

Konstanz or Lindau are both good places. If you get to Konstanz, visit Insel Mainau, a short bus ride north of the town.

There are plenty of train routes to and round the lake. You can do a complete circuit of the lake by train, changing trains a few times. There are also many ferry routes which cross the lake.

Konstanz has 2 trains per hour from Zürich airport. Konstanz station is served by both Swiss and German Railways. There are also at least on train connection per hour from Zürich to Lindau.

See www.sbb.ch or www.bahn.de for train times and prices.
The official tourist websites appear to be: http://www.bodensee.de/ or www.bodensee.eu/

P.S., in case you didn't know, the lake is called Bodensee in German.

Joan, thanks very much for the reply. Überlingen looks very nice. I'm going to check it out further.

Chris, thanks very much for the reply. My wife and I were looking at pictures of Insel Mainau Island last night and that's one place we definitely want to see. It appears that Konstanz would be the easiest place to reach by train from Zurich and it also has easy access to Insel Mainau. Hard to choose because there are so many excellent choices.

Oh, and thanks for the links and the heads up on the name. I'll check out the links.

I had one other question about Lindau and Konstanz. It appears to me from the images I can see on the Internet that the view from Lindau (or generally anywhere on the southwestern part of the lake) may be better than the view from around Konstanz. Specifically, Lindau appears to have a better view of the mountains. Any thoughts on this would also be very helpful and much appreciated.

Randall, yes correct.
The Alps are south of the lake, in Switzerland. You can see them from the north shore of the lake, but from the south shore (including Konstanz), you are looking the wrong way!

I have stayed in Lindau several times over the past 8 years; the harbor and the view across the lake are awesome. I've also stayed in Konstanz and Immenstad. Trains are good, and also good is renting a bike to travel between some of the small towns along the lake. I biked around the lake clockwise. Was a wonderful adventure. Also interesting was the town of Stein am Rhein, west of Konstanz.