Here are some tips for enjoying the best of Rome without spending a cent:
1) The Vatican museums are free to enter on the last Sunday of every month. make sure you get there really early to avoid having to wait hours in line though.
2) Rome is full of fantastic parks and churches which are all free to enter. Even small lesser know churches house some lovely little surprises and most of the parks offer amazing views of the city and are full of fountains and other monuments.
3) The Roman Forum is free to enter. Enter behind the Vittorio Emanuelle monument in Piazza Venezia and take a stroll through the ruins in the Roman Forum whilst heading towards the Coloseum. The museum across the road from the Forum is free to enter and has some great things on display from back in Roman times and follows the history of the Roman Empire. It is also free to enter.
4) In the summer months of June and July Rome is full of free concerts. The biggest nomrally take place in Circo Massimo, Piazza San Giovanni and in front of the Coloseum. Big name stars offering free concerts, what more could you want?