Rome is a walking city with a few hills. Three long walks show most of it, with plenty of coffee stops.
Getting there
Trains from the airport reach Termini in half an hour.
Where to stay
Monti or Trastevere, both walkable and lively at night.
What I would do differently
I would book the Colosseum for the first slot of the day.
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Prices in this guide are what I paid. They move every season, so use them as a rough guide.