Is it cheaper to fly to Naples and take the train from there? How is the Euro rail experience from London to Italy? Would there be a lot of train transfers? Also, if anyone has a recommendation for what we should do in either area please let us know!
A good resource for checking train connections is http://www.bahn.de/. There are often good deals on Europe's high speed trains, though you might only get a good deal to the south of France, not all the way into Italy.
It will be hard to beat a budget flight. Obviously on journey time but also on cost. Book in time and your return journey can be as little as 40 pounds with Easyjet. Still, on http://www.seat61.com/Italy.htm you can see that, apparently, you can get a roundtrip by train from London to Italy for as little as 115 pounds. It's a long journey, but it's a good price.
I would personally recommend taking the train - it means you can see the landscape change as you travel south, rather than bypassing it in the plane (also taking the train is lower carbon so better for the environment).
You can take a train from London to Paris, Paris to Milan, and then a night train from Milan to Naples, arriving at Naples in the morning:
(Disclaimer - I run loco2.com)
Unfortunately this option is likely to be much more expensive than flying. This makes me very sad, but is something that unfortunately we don't have much control over at Loco2.
You might be able to find cheaper options than Loco2 for the Paris-Milan leg of the journey at Thello.com (this site is only in French), and for the Milan-Naples leg of the journey at TrenItalia.com (this site is in English but can be quite confusing unfortunately).
You could also consider splitting the journey and stopping off in Paris and Milan. This is unlikely to affect the price much, and would make for a great adventure, but obviously you'd need more time.
I think it's worth adding detail for the Naples - Amalfi coast part of the journey. I'm assuming you're going to Sorrento, as that is the route I am familiar with.
If you do get the train to Naples, you'll want to change at Napoli Centrale station for the Circumvesuviana train to Sorrento, which is the end of the line. Pompeii (Pompei Scavi station for the Roman rather than modern town) is on the way, but you'll want most of a day for that.
If you fly to Naples airport, you can either:
get a direct bus to Sorrento. While very scenic, the coastal road is very narrow and winding in places, and can get very busy (e.g on a Friday night) which can make for a frustrating and uncomfortable journey.
get the 'Alibus' to Napoli Centrale train station and get the Circumvesuviana train from there - no train station at the airport. Probably not worth the bother, unless you want to spend some time in Naples, e.g, to visit the Naples National Archaeological Museum (Museo Archeologico Nazionale di Napoli) which has a lot of sculpture from Pompeii and Herculaneum (there are very few artifacts at Pompeii).
stay on the Alibus to the port and get a fast catamaran to Sorrento from there. Very quick, fantastic views of Vesuvius and the coast (I spent the whole trip standing on deck), relatively expensive. Arriving at Sorrento's port is quite spectacular, although it's a very steep walk up to the town, you'd pretty much need to take a bus or taxi if you have a suitcase.
I would personally recommend going by flight, so you can appreciate your time there (it's much quicker). Also, while trains used to be cheaper, there really isn't much difference, if any, when there is a deal on. There are normally some cancellation offers at http://www.directline-holidays.co.uk/