The problem with taking trains in Switzerland is that a lot of the time they are in tunnels! There is a beautiful train route that goes near the Matterhorn. Once you get to the border of Italy, you will go through Milan, Genoa, and then along the coast to Pisa and Florence. I would absolutely, without a doubt, stop in Genoa. But you'd need at least a full day there. There are two train stations. The main one (Principe) is in a not-so-good part of town and if you let that influence your day, you'd be missing out on a beautiful city. The other station (can't remember the name off-hand) is closer to the main street (Via XX Septembre) and is a nice street for shopping. Christopher Columbus's house is naer there as well. You can walk inside the walled part of the medieval city and walk down the beautiful passageways. But even more beautiful is to take a bus to Nervi and walk along the cliffs there, have lunch in a restaurant overlooking the Mediterranean or go to one of the outdoor markets or the huge park there. Most people there can help direct you to the bus - and they're VERY easy. Also, if you're going to be in Genoa, just ask anyone for "Corsa Italia" - it's a boardwalk that runs along the coast and is full of people all the time - it ends on the east side in Boccadasse, a small fishing village that is breathtaking. On the way, you'll pass Gelatialia - THE BEST ice cream (gelato) in Italy!
If you stop in Pisa, it is a nice little town, but not much to do there beside the Leaning Tower - and make sure you get reservations BEFORE you get there if you want to go up in it as they only allow a certain number inside at a time. And wear good walking shoes - the train station is on the other side of the town!
There are other places to visit as well - such as Cinque Terre (vineyards overlooking the Mediterranean, but VERY hilly and lots of walking), Carrara (white marble quarries) and some other small villages around there.