I really recommend staying inside the historic center in Florence. The city is meant to be walked. You can get a cab, but remember that you'll need to call one - you generally cannot hail cabs on the street. There may be a few taxi stands, but I don't remember off the top of my head where they are. Also, the meter starts running from dispatch, not when you get in the car.

If basic is ok, check AirBNB and see if you can get a small apartment. Or look at well-reviewed hotels in the 2 star range. The star rating has nothing to do with quality in Italy, just services offered. Basic accommodations can be cheaper, especially since you're not going in June/July.

Once you firm it up, I'll help you with places to eat in Florence.

All that said, I've posted it in other threads, but my first trip to Italy was 12 days. We did Rome, Venice, Florence, and I felt it was too busy. I'd really suggest you cut it back to Rome and either Florence or Venice, depending on which interests you more. Perche will know better, but I think you still have cruise ships coming to Venice in September. If it were me, I'd do Rome and Florence and save Venice for a winter trip. If for some reason you're bored in Florence on the 5th day, you can day trip out to Lucca or Siena; or with Rome, Ostia Antica. But I doubt it'll come to that if you haven't been to these cities before.