There is not much to see or do here, except for a PH (a kind of university just for teachers), a FH (rather technical university) and the famous Basilica there is nothing. There is not even a motorway.
South or North towards Friedrichshafen, Konstanz, Ulm
To get out of Weingarten you have to go towards the Burger King. To get there you follow the Waldseeerstraße towards the intersection at 'Domicil' and then turn left (if you come from Ravensburg) or right (if you come from Baienfurt). Just follow the street Niederbiegerstraße until you get to the next intersection with two traffic lights. Behind it there is a small strip of grass where you can stand, the cars can stop at a lay-by just behind it (see map).
If you want to travel south carry a sign with FN (for Friedrichshafen) or KN (for Konstanz), if you want to go north it is best to ask for a ride towards Ulm. If you want to go towards RT (Reutlingen) write some small town on the way an you sign (Saulgau, Riedlingen, Zwiefalten).
Usually it doesn't take very long to get a lift, even if you're two is shouldn't take longer than half an hour.