Last year we flew into Milan and drove to Abbadia Lariana ( or something like that ) which is on a train line just north of Lecco. There was a nearby crag I think called Discoteca which wasn't bad. But pretty much we had to drive to most crags. Used the trains for sightseeing days into the cities.
The trouble with crags close to public transport is they are very polished. Arco is another major climbing centre, but the popular closest crags were horrible I thought. It took about 5 days before we realised this and went to some good crags near Sarche, and up in the hills to the north. We also spent 2 half days at Ceredo, and based ourselves at Ai, also on the rail line north of Verona. This was handy for a trip to Venice, but still needed a car to get to the crag which was up a large hill about 20 min away.
Also spent time around Finale Ligure, which had some accessible crags. Free bikes came where we stayed, though again it was quite hilly. There is also a huge climbing guidebook to the Pennavaire valley to the west. Did some nice climbs around Toirano, but again cars are very handy. You may have more luck in the Dolomites. It was too early in the season when we were there (may). Crete was great fun exploring near Kapetaniana and plakias and Agio tfearano? but again car essential. Can't go past Kalymnos. Rhodes also has some crags. I think getting a car is the theme here.....