We're just back from that area, we definitely would recommend a trip to Lake Garda. Lasize and Sirmione are lovely old towns in easy reach of Verona, Limone and Riva are nice too but you would probably need to use the ferry to get to them.
I wouldn't bother with the cable car at Malcesine, the queues are crazy and it isn't that amazing, the cable car on the pordoi pass though is absolutely incredible so if you can arrange a trip there or hire a car I would recommend that, the south tyrol is so different to the rest of Ital it is really worth a visit.
The Zeni wine museum at Bardolino is a good place to visit for a bit of wine tasting that you don't need to book. Although there are other places that are nice too this is free and interesting with fairly cheap wine available to buy.
Bassano del grappa is worth a visit, go to the Poli distillery as well as seeing the bridge.
We visited quite a few smaller towns for a walk round, Parma, Castelfranco, Arco, but I am not sure how easy they would be to get to without a car.
Beautiful area though, you will have a fab time!