You have only 8 or 9 nights - 7 or 8 full days plus about 1/2 day on the 4th when you're jetlagged and maybe sleep-deprived. Your return flight from Rome will probably leave in the morning, so you can't count departure day. Don't over schedule. Leave enough time for Rome - you'll use about 1/2 day just to get there. 3 nights means only 2.5 days of sightseeing.
The Cinque Terre are very pretty when the sun is out. In early March, some of the trails may not be fit because of winter damage not yet repaired. The nicest views are from the water but the ferries don't run in the winter. Even if you have good weather, sunset's around 6.15 pm leaving you with long evenings and nothing to do, probably not much choice of restaurants then either. And you'll use up way too much time getting there from Florence and more going on to Rome.
I was in Italy this year in late March to early April. The Tuscan scenery wasn't much to look at - the vineyards were just rows and rows of black skeletons of the vines and a lot of the trees were just beginning to open their leaves. You'll be there 2-3 weeks earlier.
Embrace the benefits of late winter - the beautiful buildings won't be obscured by trees. Most tourist sights won't be crowded. It gets dark early, so stay in places that will be interesting on those long evenings. Rome is beautiful at night. Little towns - not so much.
Bologna is a university town and a rail hub. Restaurants and cafes will be open, it has an interesting historic center. You can day trip to other towns to explore - Ferrara, Modena, Parma, Padua. It's about 1/2 an hour farther from Rome by train than Florence. Go straight there and day trip to Florence instead of changing hotels.
Consider Orvieto for 1-2 nights. It is a hill town with interesting sights, a beautiful cathedral and nice views of the countryside. It's in Umbria, but there's no difference for tourists between that and Tuscany. It's about 3 hours by train from Bologna, 1-1/2 hours from Rome.