I want to travel from Rome to Florence and maybe stay at a country place very close to Florence. Can I travel by train or bus, and not have to rent a car?
Sure you can and It depends where the country place is
The bus and train system are extensive and you can travel to most places with one or the other or a combination. Once there you may need a taxi to get to your final destination. Generally agriturismos are not close to public transit.
Train to Florence and then rent the car there if staying out in countryside in an agriturismo.
I'm curious, does Uber or Lyft operate in Italy, and would you consider this an option?
another thing: find the place where you plan to stay and consider the local hosts/owners will have the best suggestions on your transportation options and ask them;
Thank you all for the advice!