We went to Rome in October. Accommodation isn't cheap, but the further out of town you stay the cheaper it is. Rome is big, so even if you stay somewhere central you will still do a lot of walking and/or use public transport. Their metro system only has two lines and not everywhere is convenient for the metro.
We stayed in a hotel next to the French embassy, so it was quiet at night, but convenient for some nice places to eat. It was about a 10 - 15 minute walk to the nearest metro station. I didn't feel confident enough to use the buses - I never do in a strange city, but the metro was really easy to use, you just buy tickets at tobacconists and other local shops.
We did the Vatican, Colosseum, the Parthenon, the Trevi fountain (under repair when we went, the Spanish steps (under repair when we went) Palatine Hill the Forum and umpteen churches.
Don't even think of driving in Rome. The traffic is bad and there is nowhere to park.
Tips - Rome has a bad reputation for pick-pockets especially on public transport. We never had any problems because we were very careful and didn't look vulnerable so we were left alone. I was staggered at the number of tourists who kept their phones in the back pockets of jeans, so I expect a number of phones will have been stolen.
The street hawkers were selling selfie sticks everywhere, but they leave you alone if you aren't interested.
We didn't find eating out any more expensive than eating out at home.
Take comfortable shoes. As you are going in February they will need to be waterproof.
Be prepared for how crowded the Vatican is. I would book timed tickets online before you go. I have to admit that after the very ornate and colourful galleries that you walk through before you get to the Sistine chapel the chapel itself is rather underwhelming in comparison. DD said she was bored with paintings of fat babies by the time we had finished. My favourite sites were Palatine Hill and the Parthenon which is astonishing when you consider just how old it is.
I loved Rome and would happily go back. We only had four nights there and didn't see everything.