we took our 2 sons and stayed at hotel White, perfect location, we walked everywhere except the vatican, when we got a day ticket for the open top bus, did a tour - got off at Vatican/Sistine Chapel had lunch - got back on bus and carried on the tour!
Can thoroughly recommend Angel Tours, as small group fun guides. We booked them for the Colosseum to make it come alive for the boys and the girl we got was excellent, she was irish and lived in Rome. They keep their groups small, I didn't think it was pricey. There were 10 of us, we met outside the tube next to the Colosseum, and got he tickets for for us, so we didn't have to queue, normal queue in April was about 3 hrs long, but there is a separate queue for groups.
One of my highlights was getting up early-ish and walking. At 8am we had the Trevi fountain to ourselves.
The Hotel White is 2 min walk to Trevi.
I can't remember if I booked through BA city breaks or whether I did the Hotel and then the flights separately.