The Coliseum is amazing, as is the Forum. It's worth reading at least a wee bit about ancient Rome before going though, rather than wander around in total ignorance like a lot of the folk you see there.
The Pantheon is an incredible building, especially if you can get in when it's quiet. And there's a tomb made entirely of human bones that is worth a trip (begins with a C I'm sure). The Catacombs outside the city are good, and Hadrian's villa is worth the trip.
Personally I wouldn't bother with the Vatican. For a church it is obscene how much has been spent in the place. I found it repugnant (though I should confess that being brought up in a strongly Protestant part of Scotland that my judgement may have been coloured by a built in bigotry!).
Don't eat/drink in the main squares unless you want to be ripped off. Trastevere has the best food. And as folk say, watch your wallet!!
On the whole though, Rome is one of the best cities I've ever been to. We've been back several times.