Remember to have some kind of shawl/ cardi in your bag if you want to pop into a church (and especially for the vatican) They won't let you in if your shoulders/ knees are showing. And remember the Pantheon is a church now so the same rules apply there.
Book tickets online to go and see the Domus Aurea (Nero's golden palace) It is amazing - but you have to pre book.
And if you're a cat person the Torre Argentina cat sanctuary is lovely - you can go in, but you don't have to - it's in the middle of the road so you can just look over the barrier. The sanctuary is made up of loads of ancient ruins - including the theatre and portico of Pompey which is where Julius Caesar was assassinated. And there's a cat just lying there - no clue as to what happened!
Definitely do anything you can to avoid Vatican/ Colosseum queues - I'm afraid they are places I haven't been inside since my first visit. The outsides are good enough.
Definitely go in the forum and palatine though - they never get old.
The Spanish steps are nice - but busy.
Keep your eyes open as you wander around because there are just ruins and bits of statue everywhere, and it's wonderful. (there's a giant foot on a plinth I think near the Pantheon - nothing else of the statue, just the foot)
The Mamertine prison is a great little visit - St. Peter's face is imprinted in the wall where he bashed it there (I mean - it's clearly just a bit of broken wall, but they say it's his face and have put a grille over it to protect it. And he was really there once upon a time)
Castel St. Angelo is gorgeous - and the cafe (expensive) looks out over the pope's secret tunnel and onto the Vatican (worth the expense for the view)
The Atlas Obscura - on the Aventine, is a key hole which you can peer through and get a perfect view of the Vatican and it's gardens.
The Lateran church has the original doors from the Roman Senate (the ones on the actual senate building are copies) and the heads of Peter and Paul. And across the road are the steps Jesus walked up to get to Pontius Pilate's house (obviously not in situ - someone brought them to Rome) Well worth watching people climb up them on their knees.
If you want to go a bit further out than the centre - there are some brilliant catacombs along the Appian Way that you can visit.
Take a book everywhere you go, so you can just sit down in some gorgeous spot and read a chapter and enjoy being there.
If anyone tries to sell you a ticket to a concert taking place in a church that evening jump at the chance.
Count the number of nuns and monks you see. (monks are rare beasts but nuns are flibberty gibbets)
laugh at all the people in their tour groups - looking miserable and bored in the most wonderful city on earth.
And make sure you throw a coin in the Trevi fountain so you know you're going back.
It is the most wonderful, beautiful place in the world.