Rome's quite compact; you can walk between the Trevi fountain and the Spanish steps and the Colisseum and Vatican City very easily if you're reasonably fit. The main airport is the Fiumicino Airport though, and you ain't going to walk from there to the city, trust me, you'll be in a taxi for 30, maybe 45, minutes depending on the traffic.
Stay at one of the hotels near the side of the Vatican City, and you'll be near the main escape to the airport and also well placed for walking the sites. (Although you could shell out and take an open-topped bus, and just get on and off when you like, but I think it's about 20 Euros these days so you might not want to pay).
Re: Hotels, I'd recommend the Best Western Spring House. It's cheap and cheerful but just refurbished, so pleasant. (I was last there in February).
http://en.venere.com/hotels_rome/vatikan…
Get over there, it's great.
Watch out for lunatics in small cars asking you for directions and stuff; they WILL try and rip you off.
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