Most day trips using public transportation are single destinations. Siena, best by bus, is well worth a full day. I havne't been to San Gimignano - yet - which is also a one-town visit by bus. Others here have said you can visit Pisa and Lucca in one day by train. I believe that's if you just want to visit the cathedral campus. Besides the tower, the cathedral is beautiful and the baptistry is exquisite. There's certainly enough to see for a full day in Bologna, by train. Those are the main day trips.
The Cinque Terre are too far to make a reasonable day trip and Portofino is even farther.
If you want to see other towns, consider a few nights in Bologna. It's a rail hub, so Parma, Modena, Ferrara, Padua and even Ravenna with its marvelous mosaics are good day trips.
Christine has noted that you'd want a car for an agritourism stay. That's good if you want to visit wineries, but note that the legal alcohol limit for driving is only .05% while in the US it's .08%. Hill town visits are problematic because parking is very limited and driving in Tuscany can be challenging on back roads.