Rome to Londaon

It is a very long trip. Go to bahn.ge and plug in your dates. Depending on a couple of factor 12 to 16 hours. This is case where a discount airlines makes sense.

If I choose a discount airline, are there "good ones" and "not so good"?

All budget airlines have strict limits on luggage numbers, size, and weight. If you exceed any of them, you'll have to pay a fee. All budget airlines have strict check-in deadlines. If you miss the deadline, your ticket will be worthless and you'll have to buy a new ticket at full fare on a later flight. That said, if you pack light and follow the rules, flying on a budget airline can be a good deal. Ryanair flies Ciampino-Stansted. Both are outlying airports so you'll need extra time and expense to get to them. EasyJet flies Fiumicino-Gatwick. If their flight times match your needs, go with them. However, don't rule out British Airways from Fiumicino to either Heathrow or Gatwick. Their fares can be competitive with the budget airlines on the Rome-London route.

I would look at Easyjet. And as noted, read about baggage charges( best to pay online upfront) and limits. Rome to London would take a long time, and likely will cost same or more by train. If you are booking early enough you can get amazing rates ,, even with the adds on fees for luggage.

I think that Frank meant bahn.de not bahn.ge. For the best overview of the full journey have a look at this article by the Man in Seat 61. The link is not working so the url is (copy and paste the entire next line)
http://www.seat61.com/Italy.htm#London to Bologna, Florence, Rome, Naples It is from London to Rome, but as you read and scroll down you will find the information from Rome to London. There are 3 basic routes by train and he goes into each one.

I'm SO new to all of this. Thanks to everyone who has responded to my question. What is it like going from Heathrow to Gatwick? a hassle? easy? time involved? Let me know.

Theresa , it would be helpful if you gave more information, when and why would you be travelling between airports for instance?