Speaking for myself I would rent a car for visiting the chateaux. Put anyway Chenonceaux on the list and even it´s just only to see that impressive outside Chambord (inside it´s a bit empty) as well. Amboise is interesting also as the place where Leonardo da Vinci is burried and the mansion Clos-Lucé where he lived at the end of his life and died there too. For the gardens as well you can think about Chateau de Villandry.
It´s also fun to drive along the Loire river and follow other touristic itineraries and visiting places like for instance Loches. The chateaux / places are so much spread out I think public transport will be too much time consuming and will limit flexibility if you like to change plans (surely can happen) on the spot.
A bit of background info about the place you are going to see can make the visit more interesting like for Chenonceaux the intruiging rivalry between Dianne de Poitiers and Catherine de’ Medici. Just reading some wikipedia articles can be helpful enough.
You can´t go really wrong with Amboise as base I guess, indeed cute, has a good atmosphere and located a bit central. Have no idea about public transport as I used my car. And no personal experience with joining a tour, but support the idea to consider it as an option.
