You'll want to use trains to move around Italy. Between the major cities they are a lot faster than a rental car, and they eliminate a many potential car-related costs (including receipt of very expensive traffic tickets months after you return home). The August rail schedule isn't posted yet, but you can take a look at the first week of June on the Trenitalia website and get a very good idea of what your options will look like. You'll need to use the Italian spellings of the city names: Roma, Firenze, Venezia. You can use Vernazza for the Cinque Terre.
If you're in a position to lock down your travel dates and times by the time August tickets go on sale (not sure when that will be, but probably not very soon), you'll be able to snag tickets at very good fares, probably similar to what you see today for early June. If you want or need to wait to buy your tickets closer to your actual travel time, the ticket prices will most likely be much higher. To get an idea of what those might look like, check on tomorrow's trains.
Most of us find that we are pretty brain-dead after a mostly-sleepless overnight flight, plus there's the jetlag. The usual recommendation is to walk around outdoors on your first day. Some people find it helpful to take a 1-1/2 or 2-hour nap in the afternoon, but set an alarm, or you may sleep straight through till the wee hours of the morning.
So that first day is a good time to do a Rick Steves walking tour. It is not a good idea to pre-pay for any sort of activity on that first day, because you may be in no condition to enjoy it (or remember it). You can enjoy walking around Piazza Navona, see the Trevi fountain, go to the (free) Pantheon, maybe pop into a church or two. Just enjoy being in Rome.
