You could easily spend several days around Rome. I think the Roman Youth Hostile was a bit out of the way near an Olympic Stadium, and the bike was nice to get around the city, although the glass was hard on my sewups (a few decades ago). I remember a nice ride out a Roman road outside of the city near Rome, but don't remember the name.

About a day's ride from Rome, is Terni, which also has a small area of Roman ruins which I thought was quite spectacular, with an actual segment of original Roman cobbles with very very deep wheel ruts.

If you wish to hit the big tourist areas, you may find it advantageous to also use the train. Look up the current rules. I was kicked off the train once for taking my bike aboard, but things may have changed over time.

Assisi might be a smaller town a bit off the beaten path, and perhaps important with respect to our recent Pope's namesake.

Pisa and the Piazza Dei Miracoli is a wonderful place to visit for a day, but it would be a long ride from Rome and back.

If you are in Italy during the summer, consider the Palio di Siena.

Cinque Terre is also a magical place, although perhaps not the best destination for biking. There is a path between the 5 towns, but primarily a walking path.

And, don't forget Venice.

Depending on what kind of cycling you like, the Po Valley is FLAT and very bicycle friendly, from Milano to Bolognia, and up to Venice.

I had fun touring many of the little castles and walled cities around Italy.