Take your bike!
Near the beginning of April, my son and I did a multi-day bike ride along the Rhine (started in Chur). That section after Schaffhausen was one of my favorites.
Take the train and park the bike at the train station (downstairs in the bike parking, you can pay 1 CHF to leave your bike in a secure location). Walk around Schaffhausen; a couple of hours is more than enough to see the old town. Then get your bikes and look for the bike path #2 (Rheinradweg 2 or something, you can't miss it). As you are approaching the Rheinfall, look for Routes 77 and 752 to go visit the falls. Cross the border to Germany for a bit and then cross back into Switzerland at Rheinau and rejoin Route 2 again. Bike all the way to Eglisau and then catch a train back to Zurich.
Here's a map:
Bike Route 2