Transportation

I presume you would be visiting Rome either at the start or at the end of your trip (the end is better) since you are flying to (and probably from) Rome. To visit Rome as a day trip from Assisi would be crazy considering you'll have to be there for your flights. Assisi is not that close to Rome.
To visit Siena, definitely drive and park at the Fortress next to the stadium. Just follow signs to the stadio (often symbolized in road signs by the image of a soccer ball). Going to Siena from Assisi via train is cumbersome and long.
Arezzo is also easy and fast by car. Alternatively you could go by train from Terontola or Assisi itself. Similar travel time, but car would add to flexibility because you could also easily visit Cortona along the way and easily kill both in a day.
For Florence it's a toss. You could either drive or take the train from Terontola or even Assisi. ETA would be similar, once everything is taken into consideration (but some trains are faster than driving for sure). The car however would give you the flexibility to linger in the city later, since you would not be tied to train schedules. Last Regionale train to Assisi departs Florence at 8:13pm, last Intercity train departs at 9:42pm.
From Assisi I also recommend you visit beautiful Anghiari. Just take the freeway north yo Cesena and exit at Sansepolcro. Easy.

Some great advice. As far as Rome, we were just planning on visiting it at least 2 days taking day trips from Assisi. We were thinking of driving the car and then parking it somewhere on outskirts of Rome. We would do the same for Florence, taking a couple of day trips. The other 3 days would be just exploring around Tuscany.

Where would be a good place to park the car in Rome and Florence?

Would it be better and faster to just take the train to Rome since it is farther?

You should go to Rome a couple of days earlier, spend a couple days there before flying home. It's a long trip from Assisi on a day trip, tiring and not cheap. It's 2.5 hours by either car or train. And that doesn't include the time to park your car in some parking lot. The train is faster (as much as half hour faster), especially considering that you will encounter a lot of traffic on the way out of Rome unless you leave very late at night. The train is not affected by traffic, doesn't need to be parked and leaves you downtown.

For Florence the train is probably just as fast or faster. But if you want to drive, park at either the Oltrarno lot on via Della Calza (behind Piazzale Di Porta Romana) or on Piazza Beccaria. For the former you would exit the freeway at FIRENZE IMPRUNETA for the latter you would exit an exit earlier at FIRENZE SUD. Use Google maps for directions.

Thanks so much for your help.

But I just saw this on your Rome post:

"What are a few moderately priced good hotels that are close to Termini Station. My husband has heart failure and cannot walk a long distance."

Given that, I'd take the train into Florence as it'll deliver you directly to the city center.

I'd also point out that Assisi has almost no flat areas. Just about anywhere you want to go is up and down walking; some of the streets are staircases, because they are so steep. If your husband cannot handle more than a couple flights of stairs, Assisi may be too challenging for him.