Brian,
A few points to mention.....
For driving in Italy, each driver will require the compulsory International Driver's Permit, which is used in conjunction with your home D.L. An I.D.P. is easily obtained at any AAA office, and is valid for one year.
The traffic restrictions that Roberto mentioned are referred to as Zona Traffico Limitato, and I've found that most towns have them so be sure to do some research on that. You may find it helpful to have a look at THIS website. In some cases they're enforced by automated cameras (Florence has extensive coverage with cameras), while other towns rely on the "cop on the beat" to issue tickets. The restrictions often apply during specific hours.
When visiting Cortona, there's a large car park (On Viale Battisti, free as I recall) just below Piazza Garibaldi. From there you can take two Escalators up to the town. Assisi (Umbria) also has ZTL areas, and was told by a local Guide that cameras will be installed there in the near future.
Unless your hotel in Siena offers free parking, you'll have to pay to use the public lots. If you just want a quick look at Siena, taking a day trip by Bus from Florence as Roberto suggested is a good option (DON'T forget to validate tickets!). I usually prefer to spend at least a couple of days in Siena as it's a beautiful city with lots of history. You'll be there before the Palio, so hopefully it won't be too crowded.
In late May/early June, I would highly recommend pre-booking all accommodations. That's in the spring shoulder season, and I've found it can be VERY busy then. While you may be able to find accommodations on short notice, it will take time to find them and you may have to settle for something that's either in a less-than-desirable area or at a price point that's above your budget. Pre-booking is the best way to get exactly what you want.
Happy travels!