You should check it out before it goes underwater forever. It's a short direct train ride away from Verona. You definitely don't need to stay the night, it's small enough that you can just do a day trip. If you're planning on taking a gondola ride, make sure there is a sign with prices at the dock.
A personal favorite. Make the time to visit at least Riomaggiore, Vernazza or Manarola. You can take the train to La Spezia and from there take the Cinque Terre Express train that goes through all 5 cities.
Rome is huge but again no car needed, between taxis, public transportation and your own two feet you'll be golden. For places to stay, I suggest staying around the Spanish Steps or Trastevere. Anything around the Vatican is a bit far from what you want to see. A few places I highly recommend (we have more if you want more details!): Amatricianella - best authentic Roman cuisine around. Il Giardino degli Aranci (The orange garden) - And it's famous keyhole view to the Vatican. The Roman Forum - It's out in the open but you'll need tickets to enter. Villa Borghese gardens - Next to the Spanish Steps. Castel Sant'Angelo and the Vatican Museums.
Those are a little tricky to get to with just public transportation. You may want to rent your own car to get here.