In reply to potted shrimp:
A car helps a lot BUT isn't needed. Friends have definitely managed fine on the buses.
Areas with the highest concentrations of routes within easy reach are those near to the Falzerago Pass probably. Cortina jumps to mind with buses in a few directions and a good number of shops, campsites and other amenities. Corvara is similar for that respect but a bit quieter and a bit more awkward to get to.
Cortina has two/three via ferrata you can get on from by ski lifts in the town centre: sci club 18 is a reasonably short day but a lot of fun & steep to make up. If you stay on the lift past the climb you can get up higher to at least one walk that my parents have done and a sport climbing area. There are also a couple of via ferrata and potential walks off the lift up Tofana di Mezzo (these normally make for pretty big days and are pretty hefty).
A bus up the Falzerago pass takes no time really and puts you at the lift for the tomaselli route and also the base of the alpini brigade route. Going the other way I think there's a bus to the cristallo lift but I really can't be sure (easier ferrata and some walking). There's good climbing on the Falzerago and good walking too. For sport I'd recommend the 5 tori (bus stops at lift station) as it's a brill location and the climbing is pretty fun. There are some huge routes you can tick off around the Falzerago too e.g. Up Tofana di Rozzes. At the very top of the pass there is sport and trad climbing accessible from the bus stop.
Corvara also has a couple of pretty accessible ferrata and plenty of walking. There is plenty of climbing around and about too. I've not climbed there in an age but the Tridentina ferrata sticks in my memeory for cabled fun.
If you get one book for climbing/ ferrata the rock fax one is probably where your money is best spent. It's not the best written, nor does it give the best descriptions, but it does cover a very good mix of routes! Walking wise there's a nice little cicerone thing I've borrowed from my parents before now. Those would give you a good idea where things are