Florence is small, with car it is better to park it and forget about it!
Ciao Marge and welcome to the forum!!
Completely agree with you, if you will have a car for your entire stay while you're in Florence you want a place where you can park it safely and forget about it while you visit the city. You want only apartments or would small hotels/B&Bs also work out as well?
Here are some I'd suggest with parking in the center or just outside of the center "perimeter", still very close to walk into.... oh, Florence in general is really safe - the greatest danger is traffic! but once you're in the center, it is mostly pedestrian areas with very limited traffic and so you'll be fine.
Ok these are within the center and with parking still outside of the limited traffic area (ZTL) - you definitely don't want to end up in the ZTL since you won't have a permit, so make sure to follow their instructions on how to get to them exactly so that you don't run the danger of the ZTL area and fines.
In the center - Apartments and small hotels:
Pergola -
http://www.florenceaccommodation.com/apartment-pergola.html - has a private garage spot, just ask when you book to get this too
Guelfa -
http://www.florenceaccommodation.com/apartment-guelfa.html - same owner as above, so same applies...
Oltrarno -
http://www.florenceaccommodation.com/oltrarno-apartment.html - paid street parking nearby, outside of the ZTL marked area.
Hotel Principe -
http://www.florenceaccommodation.com/hotel-principe.html - a small family run hotel with garden really close to the American consulate (so very safe and very central). The hotel has parking spaces for guests behind the back garden.
Just outside the center - all within walking distance from the city's main sights.
Palagetto is a small B&B with its own parking spots -
http://www.florenceaccommodation.com/il-palagetto.html
Villa Antea also is a small but very elegant B&B just outside the historical center perimeter - also has its own garden parking area:
http://www.florenceaccommodation.com/villa-antea.html
Johanna II is another small B&B with its own private garden and parking area -
http://www.florenceaccommodation.com/johanna-ii.html
Here are some more options, including several apartments further away from the center or just outside of Florence, if you want to be more in the "countryside" rather than the city:
http://www.florenceaccommodation.com/florence-accommodation-with-parking/
Let me know if you need more specific suggestions - if you see something you like, tell me so I know what sort of places you're looking for and can point out some that I think you'd also like.