i'm taking a very similar trip next week. Going to Nice for 5 nights then flying to Naples for 3 days in the amalfi coast and then 3 days in Rome. I will grant you i've already been to florence and venice a bunch of times, but it is way way cheaper to fly around from nice into italy. once you're in italy - take the train. 10 days is not a lot if you haven't been there.

i'd do something like this:

2-3 days in Nice. visit monaco/vence(car needed)/antibes/cannes/etc

Fly to Venice (there used to be an overnight train to/from nice/venice, but it got cancelled). check airfrance for deals - i am paying $2 USD (plus tax = $57) for my ticket to Naples. easyjet also had deals. so did alitalia.

stay in venice 2 nights. train to florence. stay in florence 2-3 nights. train to rome. stay in rome, and enjoy.

that's just my $0.02.

have a great trip!