I don't understand why you need to stay near the airport. Yes, it's out of town but trains runs regularly to/from the main train station in town.

I suggest a few days in the city, I prefer Roma Antica (the ancient section).

FWIW - the most direct route would be UA SAN-IAD-FCO (though I doubt this would be the cheapest). For cheapest I'd try Gate1.com or site59.com (I think it's now lastminute.com).