Italy is pretty much mountains everywhere

I would agree with Boa that Genova is a good place to go and the Portofino reserve is much dived. However, it depends on what kind of diving you like to do of course.... If travelling along that coastline (its the Ligurian Sea there BTW) its well worth crossing over into France and diving in the area around Nice and also Marseille. There is some fantastic wreck diving in that area (
please bring a medical certificate of less than a year old to dive France).
From there you can return to Italy via either the mountain pass to Cuneo or through the Mont Blanc tunnel. Both are spectacular mountain drives in stunning scenery. The passes are very high (2000m+) so please leave plenty of time after your dive. If returning via Mont Blanc you can stop just on the Italian side and take the cable car to the summit (the mountain is called Monte Bianco in Italy)
Its really an absolute must-do if you like mountains.
From Monte Bianco you drive back towards Torino through the Aoste valley. Again a stunning drive (if the air is clear - Italy is a horribly polluted country - August is the best month). The Parc de Grand Paradiso is in that valley and here you will find (if you look hard and buy a good guide book) Griffon Vultures and Ibex. The area is half French half Italian and fiercely independent. The local cheeses are incredible.
From there head towards Milano (a dirty and disappointing city - I used to live there). Just to the north of the city are the Italian lakes. These are also spectacular and offer some limited cold water diving (they are very deep [300m+ thats 1,000ft] - be careful) In the summer the banks are lined with Italians sunbathing and they park anywhere and everywhere so progress can be slow at times.
Be careful driving round the lakes particularly on Sundays. Italian drivers are all lunatics and if you are used to the 55mph limit in the US you will need to speed up a bit
Trento is very nice and the South Tyroll too. Its more German than Italian and has a very different feel to it. You might also want to cross over into Switzerland while you are there. Again the mountains are beautiful and some breathtaking lakes too.
From there across to Venezia. From there you might also want to cross over into Croatia and the spectacular Dalmatian coast. Here too is some fantastic diving. The island of Krk is worth visiting and you will find plenty of dive centres.
If you are getting a hire car at the airport the insurance should cover you for all the countries mentioned. Try to get a diesel as gasoline is very expensive in Europe. Most cars are shift unless you get a luxury model. Diesel is about $8/9 a gallon and gas about $10/12. There are road tolls too in Italy and France. For Switzerland you buy an annual pass (one year is the minimum you can buy) at about $60. Hotels are expensive in Italy but cheaper in France. The big seafront hotels in Nice are for the super-rich but out of town chain hotels (Ibis, Camponile) are about $90 a night.
I hope you have a great time. If you want any more information on the area let me know.
One last point, Europe is on holiday in August and getting hotels can be difficult. You might want to book in advance, particularly for the tourist areas (Roma, Venezia, Nice [Nizza in Italian] etc) The freeways get pretty heavy too expect long delays. The border crossing from Italy to France can take an hour or more at times just due to traffic (Europeans do not need a passport for that crossing - most the line out is the freeway tolls).
Have a really great time - hope you get to visit Mont Blanc I will remember the cable car to the top for the rest of my life.
Chris