Just got back from there for the second time. Must do's are the Vatican including the museum and Sistine Chapel (try and book a tour or at least get your tickets online to avoid the queue). If you want to go to St Peters Basilica then go early, around 8am or earlier before the queues build up. Also the Colosseum, the Forum, Trevi Fountain (currently closed for refurbishment) and the Pantheon. Everything else is a nice to do in my opinion.
It's very difficult to eat badly in Rome, but if you want to eat cheap then just buy some pizza from one of the many shops and go sit on a bench or park. I went to a couple of great restaurants; one that I'd highly recommend is Life Restaurant.
Use the underground/metro, it's cheap and easy and can save your tired feet at the end of long day.