There's a really good train connection from Gatwick to London called the Gatwick express - the trains are really nice and new and clean (on the whole) and as it says it's an express service, I think it takes about 20 mins to London from Gatwick and only stops once or twice.
I have used it before but only once - when I was in Acton last week for my FDFC one of the girls came from Crawlay and caught the Gatwick Express there and back and managed to get a seat on every journey, and for a train anywhere between London and Brighton at rush hour, that's a minor miracle!!!
Gatwick Train station is within the Airport and well signposted, then you'll need to use the underground to get to Olympia
Apparently, there is a train from Gatwick straight through to Olympia! If this is not going at the right time for you, I would definitely recommend the Gatwick Express, Trinity's right. (Going on Monday so I can't meet you, shame).
West Brompton for Earls Court (Mainline trains)
Kensington Olympia (Mainline trains)
Silverlink and South Central Trains run direct services to both these stations from Clapham Junction, Gatwick Airport, Brighton, Watford Junction, Milton Keynes, Northampton and Rugby.