Been to Italy 3 times, never on a bike but always with a eye towards the cycling possibilities.

1) Ditto skipping Naples. The worst town in Italy. Other then the region along the coast to the south, I'd stay away.

2) I've read and been told that Rome is very unfriendly to bikes and cyclists (but had a terrific time in Rome on foot - loved Rome). Thus would not plan on starting or ending my Italy trip in Rome on a bike. I'd use a train instead to get into/out of.

3) Tuscany is hilly. Other then the rare long valley, such as the Arno valley to the west of Florence and the feeder rivers, there's not a lot of flat. OTOH, it's simply a beautiful area, so if you have a triple, you'd probably have a great experience.

4) Umbria is a hair flatter then Tuscany. It has the Tiber river flowing thru the heart so is easier in that sense. Of course like most of Italy, the towns are all on the hilltops - easier to defend back in the Middle Ages etc... so you end up going uphill to the hotel, is the typical end of day.

5) I've been to both of the above regions April into May, 2 weeks in each. The weather at that time of year was 70's during the day, so shorts and short sleeve jersey. For me I actually like it cooler so March into April would be perfect at this latitude.

6) In general, it's warmer and flatter on either coast.

7) There's no riding to be done in Venice. The historic district is off limits.

In truth and with rare exception as noted above, you could throw a dart at a map of Italy and go there, and have a terrific time. My wife and I love Italy and will go back a 4th time on our next trip, even though I still want to see more of France, would like to see Ireland, would like to see the Balkans and a lot of former behind-the-iron-curtain countries that we could never go to. Italy is that special, as far as I'm concerned.