Pam,
I agree with the previous reply..... more information would be helpful.
If you only want to see Nice and sights in that area, then flying inbound to Nice would be the best way to avoid wasting time. You didn't say where you're flying from, but you may have to fly into a hub airport (CDG?) and then take a connecting flight to Nice. There are all kinds of possibilities in Northern Italy.
I would highly recommend using well planned trips by train or other public transit as it's often faster and more efficient. Travel on the high speed trains is especially fast as these travel at up to 300 kmH. Rental cars have a few potentially expensive caveats to be aware of.....
- as the previous reply mentioned, renting in one country and dropping off in another usually comes with a hefty charge.
- each driver will require the compulsory International Driver's Permit for driving in both countries. These are easily obtained at any CAA / AAA office for a small fee.
- driving into towns in Italy can be challenging due to the ZTL (limited traffic) zones. Some are enforced by automated cameras and each pass into the zones may result in a hefty fine, which you likely won't be aware of until several months after you return home.
- there are also parking costs, fuel costs, tolls on the motorways, automated speed cameras, etc.
I tend to rent cars "strategically" in cases where public transit doesn't go to the places I want to visit, or where the transit schedules don't fit with my itinerary. I like to maximize the time in each place I'm visiting, and minimize the time spent getting there.
You may find it helpful to have a look at the Rick Steves guidebooks for the areas you want to visit, as that will allow you to focus on the areas of most interest, and work out a good itinerary for the time frame you have. Europe Through The Back Door has sections on car rentals and transportation, as well as itinerary planning. You may be able to find the books at your local library.