Hello June,
The areas of interest in Florence are small and walkable. As long as you stay near the core of the city it's hard to go too wrong or be too far from the action. As people have said the Oltrarno neighborhood is across the river and less touristed but is also farther from the train station so a cab may not be a bad idea for arrival and departure with bags if stay there.
For a city with ancient streets Florence does a good job with public transportation using buses for the exterior of the city and bigger vans for interior streets. Use Google maps public transportation tab for bus schedules and pop into any "T" marked tabac shops for a bus ticket. Validate the ticket in the stamper machine on the bus. Florence is really walkable but sometimes you're at the end of your walk and the wrong end of the city and a bus helps out.
It's been too many years since I was in Rome for me to comment but I remember the bus routes being confusing because they seemed to wind around the city in loops. Google maps has done a good job in other Italian cities and there are Rome bus apps as well. I know they just added pay by credit card tapping on at least some Rome buses but this is new.
In all of Italy you're only supposed to catch cabs at the cab stands - they are outside all airports and train stations and scattered around the city. Otherwise you can use a taxi app to call taxis or ask a nearby hotel to call one for you. There is an additional charge for calling the taxi to you. I used AppTaxi successfully in Verona, Vicenza, Padova etc but there are several - check the taxi stands signs or cabs for which taxi app the recommend.
It can seem intimidating but just remember that people use them everyday to get around the city so you're just joining in for a couple of days and it's no big deal.
Have a great trip!
=Tod