Third vote for Ostia - it's brilliant!
Unless going into the Vatican is massively important to you, don't bother - St Peter's Square is magnificent, but the queues for the Basilica are HUGE and the museums are very expensive. Bear in mind you have to traipse round the Vatican museums to get to the Sistine chapel, which is tiny, dark, crowded and disappointing. Don't even try and buy food near the Vatican - the mark ups are obscene.
Top tip would be to buy the Blue guide to Rome - it goes into huge amounts of historical detail, especially on the more confusing sites e.g. the Forum, which can look like a pile of rocks if you don't know what everything is - will save you lots on tour guides and prevent confusion!
Try and get some arrancini (also known as suppli) - best food ever. Supplizio's genuinely changes my life 
For something off the beaten track, the medieval church and 1st century Mithraeum under St Clemente Basilica (really near the colosseum) are incredible - just check the opening times as they close in the middle of the day for a few hours.
The Protestant cemetery is also amazing, especially if you're a fan of the romantic poets.
Most of all just have fun! Eat lots of food, drink lots of wine - can you tell Rome is one of my favourite places!?
(Oh and pack a light raincoat - Rome's warm but gets more rain than London in an average year, so be prepared!)