Hmm... there is actually a lot of places around lake Constance, thus unless you have some priorities (nature, architecture etc.) it will be difficult to see all of them in one day. Especially counting the fact that you need to get to Basel some time (?) in the evening.
But if you ask us, then we would probably get Quer-durchs-Land-Ticket (since it is weekday) and start the journey from Fussen by RE 57507 till Kaufbeuren (9:07-10:07), change there for train ALX84144 to Lindau (10:20-12:01), take a walk around Lindau (
http://www.lindau.de/), then get a train to Friedrichshafen (run almost every 30 minutes-1 hour), from there take a bus to Meersburg (
www.meersburg.de), then a boat to Konstanz (
http://www.bsb-online.com/) or if you have time visit Uhldingen-Mühlhofen from Meersburg and then go to Konstanz, see the town. From Konstanz there are option with one change in Radolfzell to Basel Bad almost every hour.
You can also always go online and check out destinations you would like to visit, then list it here and we will try to figure out the way to manage transportation