Kind of agree that the colosseum isn't that great inside, but I think that was due to just how busy it was, and with the tour guide led groups everywhere you just can't get near anything. Has to be done though really doesn't it, and it is impressive. The ticket for this covers the Palatine Hill and Forum as well, which are definitely worth doing. I'd recommend getting your ticket at the palatine hill entrance and then the queue at the colosseum for those who've already got a ticket isn't long at all.
I definitely second the poster who said to go to the Baths of Caracalla. Absolutely amazing and much quieter than the more popular tourist spots. Not too far from the colosseum, so we did palatine hill, forum, colosseum and baths in a day.
Go into any open church you pass. Most of them don't look much from the outside, but inside they're incredible. To the point that by the time we went to the Sistine Chapel we were like 'meh, seen better'.
The Capuchin Crypt is definitely something a bit different. Still can't quite decide how I felt about that...
One thing i would have liked to have seen but didn't get round to is the so-called lasagne church. Various churches that have been built on top of each other over the centuries and you can go down to look at them all. Sounds interesting :)
One general bit of advice - there isn't a huge amount of information anywhere so some kind of tour is necessary for most things to get the most out of them. Pretty much everywhere has guided tours or audio tours you can purchase, or we just downloaded the Rick Steve's Audio Europe (or something like that) app which had loads of audio tours on it - a bit American, but tolerable.
Make sure you make time to just sit outside cafes soaking up the atmosphere as well :)