One of my best memories is the church/basilica of St Clemente, south of the coliseum. The priest who was in charge of it many, many years ago was an archaeology enthusiast, so he started excavating underneath his church!
You can go down steps to the remains of a very early Christian church, and then go down further steps underground to Roman ruins. It really bring to life how Rome is just layers and layers of history, quite literally.
The Pantheon is also a marvel.
I found the Vatican museum lovely, and worth the queue, but you have to be interested in catholic history, art and architecture.
I also recall having the impression of bits of Roman ruins practically every few streets, just everywhere all this amazing archaeology.
The coliseum is over-rated. Just look at it from the outside for free. Instead go and see El Jem in Tunisia. In fact, the very best Roman ruins are all across North Africa - better preserved in the desert and pretty much unspoiled as they are ignored by the locals, I think as they are Muslim peoples they are not into pagan remains. I wouldn't risk a trip to Algeria, but Tunisia is relatively safe.