daniellemichael,
First of all, congratulations!!!
Regarding your proposed Itinerary, I agree with Laurel that a slight change of order would be more efficient. I'd also suggest traveling Rome > Cinque Terre > Florence > Agriturismo > Venice. I would have suggested starting in Venice and ending in Rome, but as you already have tickets, that aspect is now set in stone.
You might consider waiting until you arrive in Florence to pick up your rental car, as you don't really need it except for travel to the Agriturismo. All the other locations are much easier and quicker by train. Note that for driving in Italy, each driver will require the compulsory International Driver's Permit which is used in conjunction with your home D.L. Also, you'll need to be VERY careful to avoid the dreaded Zona Traffico Limitato (limited traffic) zones that are becoming increasingly prevalent. Florence is virtually saturated with ZTL cameras, so you'll have to be careful exiting the city after you rent the car (hopefully Roberto will spot this Thread, as he can advise on the best routes to follow).
I'm assuming from your title that you don't have any lodgings booked at all for your trip? Especially for travel "this summer", it would be a really good idea to get cracking on that!!! At this late hour, it may be very challenging to find something in the Cinque Terre. Which of the five towns are you planning to stay in?
You can find good hotel suggestions in the Italy 2014 Guidebook, which lists a good range of accommodations in various price ranges and various parts of each place you'll be staying. The Guidebook also has great information on restaurants, sightseeing and transportation (you may want to pack a copy along, or download the E-book version of the book).
On the topic of "transportation", there are some potentially expensive caveats to be aware of when travelling by train or other public transit in Italy. A brief summary.....
- On Regionale trains (no reservations required), tickets MUST be validated prior to boarding the train on the day of travel. DON'T validate them any earlier than required, as they have a "shelf life" once stamped. Failure to validate may result in hefty fines, which will be collected on the spot! The same applies to Buses or Metro in Rome.
- On "premium" trains such as the Freccia or InterCity versions, reservations are compulsory and these are specific to a particular train, date and departure time. Passengers have an assigned Car no. (Carrozza) and an assigned Seat no. (Posti). You can NOT board any train, it can only be the one specified on the ticket. Again, hefty fines for those that don't have valid reservations for the train they're riding on. Using the fast trains such as the Freccia trains is often the most efficient use of travel time even if they do cost more, as they travel at up to 300 kmH (ie: Rome to Florence is only 90 minutes, and a very nice trip). Passengers can save money by pre-purchase of tickets for premium trains. You may find it helpful to have a look at the excellent Ron In Rome website.
Buon Viaggio!