Villa Adriana is a wonderful ruin and would be a fabulous day trip. If you saw the television version of Angels in America heaven of the gods was shot at the large pool there. Tivoli nearby also has the Ville d'Este which has interesting gardens with complex and beautiful fountains. Parking is easy at Hadrian's Villa; I don't know about it at Ville d'Este.
Just north of Rome are Tarquinia and Cerveteri which have huge Etruscan Tomb sites which are quite different from each other and worth a trip. There are many Etruscan sites in the area north of Rome so you could also find tombs in the hills e.g. near Sovana there are several and you can get maps to drive there in the town.
Orvieto which has pretty good parking is a good day trip. If you go there be sure to do the tour of the underground cavern network and of course the Cathedral is lovely.
Another simply amazing spot is Saturnia. Here is a link that shows the outdoor hot spring waterfall which is free to visit in a local park. No changing facilities when we were there years ago -- but worth changing by the car just to experience this amazing place. http://www.discovertuscany.com/maremma/thermal-baths-of-saturnia.html