just walk everywhere because there's so much amazing stuff to see and you don't want to miss anything. all the obvious stuff like the colosseum, the vatican/sistine chapel, the roman forum and the pantheon are all breath-taking. the roman pizza is cheap and delicious, i actually preferred to the more famous neapolitan style - thinner and crispier, less doughy, but search restaurants on trip advisor in advance because the best places were off the main tourist squares and less expensive.
one my favourite memories from the trip was going to a lazio game with the mrs, watching the mental ultras behind the goal from afar, as much as we watched the match, and getting pissed on these espresso liquor things that they sold to you at your seat.