Some ryan air flights are as low as .01 British Pounds.  You’ll have to pay taxes and fees of about 21 pounds, but the total for your flight will be less around $45 US.  If you’ve got more than 15kg of luggage to check your overweight fee may be more than your ticket.  You’ll not be using the major airports, but you should have no trouble getting to/from Stansted and Ciampiano

If the flight works for you, buy it NOW.

Yep, Ryanair are very legit. They fly an all Boeing 737 (jet) fleet, most of which they’ve purchased new within the past 6 years or so. Stansted is a ways out, maybe 45 minutes from Liverpool St, and 13 Pounds, so be sure to include that in your total transport costs. Ciampino Airport is actually closer to Rome than DaVinci Airport, which is the main international airport for Rome. Be sure to compare costs from Heathrow, Gatwick, and Luton, but I doubt you’ll do better because Ryanair’s specials are so cheap.

Which is the better london airport to use?

My favorite London airport, all things considered, is Gatwick. Check EasyJet to Rome from Gatwick – that’s often available cheaply.
But if Ryanair and Stansted were cheap enough compared to EasyJet, I’d take Ryanair and Stansted.