Rome is very walkable, don't use a taxi unless you need to get from termini to the vatican and don't want to take public transportation.
Daytrips from Rome include ostia antica, frascati (villa aldobrandini), viterbo (villa lante), ninfa (near norma), sperlonga, hadrian's villa (near tivoli) and Bomarzo (cool stuff).
However, in saying this, you are better off going up to frascati or viterbo in June/July. Its hot and unpleasant to be in Rome and places like hadrian's villa and ostia lack shade. What we have done is get up early in the AM, hit the market, do some walking and sight seeing, get a nice panini for lunch and then siesta for the afternoon.
Lots of museums have late hours in the summer, so you can do something cultural in the am and then again in the pm.
Longer trips, such as florence, pisa,siena and venice should be overnighters, you won't get alot out of a day trip. In saying this, I am making my poor parents do a day trip to Florence in two weeks. Hmm. maybe I'll reconsider.
lala