Welcome! Here are my suggestions/comments:
We always use Kemwel for rental cars in Italy, as we have found their rates better than autoeurope (at least for rentals in Italy). I even checked recently with the discount at autoeurope and our rate for this summer is still cheaper with Kemwel: http://www.kemwel.com
Rome 5/30-6/02 - Assuming you get there on the morning of the 30th, you'll have enough time to see the main highlights. Stay awake all day by staying out and using that first day to its fullest. You'll really be on sensory overload with so much to choose from, and lovely jetlag, but pick the main things you'd like to see and know you'll be back again someday.
I'm not sure if the National Museum of Rome is still offering a multi-ticket or not like they did for the Jubilee year, but it gets you in to several of their museums for one lower price - Palatine Hill, Coleseum, Cpryto Balbi, Palazzo Altemps, Baths of Diocletian and the main Palazzo Massimo (not to be missed for its frescos and mosaics).
Don't miss Blue Gelato, with several locations around the center (one just off Campo dei Fiori and another off west from the center of Piazza Navona).
Civitella d'Agliano 6/02 - If you are driving up their by car, you could take the scenic route up the Via Cassia (SS2) and stop off in Sutri, if you are interested in Etruscan stuff (actually right off Via Cassia as you approach Sutri. An ampitheater carved out of tufa rock and many caves/former burial tombs). We found Viterbo to have a nice centro storico.
Florence 6/03-6/6 - Are you staying outside town, or do your accomodations have parking? I might spend a few days driving up from Rome, stopping in Siena, San Gimingnao, the Chianti area, etc. before heading into Florence, where I would drop off my car before spending a few days in town.
San Miniato 6/6 - Do San Miniato while you still have the car. Very small, with a nice walk up to the former restored castle tower you can climb for a nice view. The town really shuts down for siesta. You could combine with Lucca, depending on the day of the week. Mondays are best spent wandering, as many museums and stores are closed.
Cinque Terre 6/7-6/9 - Having been to other coastal parts of Italy, I wasn't really that impressed with the Cinque Terre. LOTS of tourists. I think all the attention Rick Steves has given the place has really had a negative impact. However, if you want to go to do the hike, there are beautiful views. We stayed in a town north of the official area - it is now a national park of sorts - in Moneglia (just the right size town with your typical italian beach) and took the train into the area one day. If you just want a great little fishing town with a clean beach and some peace, try Camogli (but be warned there are few restaurants in town - do deli picnics), around the penninsula from Portofino. A car is pretty useless in Cinque Terre itself.
Verona 6/9 - haven't been yet, but by all reports it's a lovely city.
Venice 6/10-6/13 - Definately!! We love Venice, but you do need to get off the main drag. See the cathedral and go to the top of the campanile very early. You'd be suprised how early the lines form. We like the Doge's Palace and Torcello. Don't waste time with the archaeology museum, there are better one's elsewhere, such as Florence and Rome.
We found a nice pizzaria/trattoria down the street for San Giovanni e Paolo, I believe called La Strega (there was a rod iron witch sticking out of the building above the door) on Via Barbaria d. Tole (if you are facing the side of the church, there is a street that runs off the back of the piazza. At Calle Ospedale, keep going straight - you can see the witch from that juncture.)
Sounds like a fun trip - not too much, but enough to give you a tast of Italy! Feel free to email me for more detailed info.
Ciao, Kristy S.F.