I did a trip with my parents years ago from the south tip to the north on the east coast by car. Quite interesting however the south feels a bit neglected. There's not too many people there, everything is a bit run down and the restaurants don't have so full menus. Yet it is very nice because it isn't overrun with tourists.
The Gargano is a very nice area, it's that little tip above the "heel" when you look at the map. It's a national park with forests and small towns and it's quite different from the countryside around it. It almost felt a bit magical.
Up further north you probably want to see Florence and Rome and maybe Ravenna and then Venice of course. However plan your route in advance because changing from east to west coast isn't that easy due to the Apennine mountain range that runs north-south.
Even further north, you can visit Verona, which features the 2nd largest colloseum after the one in rome. In the summer it's used for open-air opera. If you love opera, you should really visit. It's open air, but without microphones and speakers, because the "arena" as the locals call their colloseum, has an awesome accoustic. No need to dress up much when you go for seats higher up. Some locals even bring pic nic baskets when they have seats on the top rows. Verona also has a very nice medieval old town to explore.
I think we travelled in August and we always found a nice room without any advance booking - that is when you have a car and don't mind that sometimes your hotel is a bit away from the town center.
Oh, and if you order pasta with fish or sea fruit, don't put parmesan on them or you'll look like a bloody noob
