Like most UK rail, it’s anywhere between somewhat expensive and _outrageously_ expensive, depending on exactly when you buy the ticket and how. Looks like it cost me 9.50 last time I was there, which is not _horrific_ for an airport (airport transport generally being a bit of a ripoff), but the barely-faster Gatwick Express would have been far more.
(The UK rail system is a uniquely user-hostile one, in that it seems designed to fool you into paying more than you have to.)
You go to Gatwick airport you take a train, which is very good (although pricey), same for most of the main airports around London.
You should definitely avoid crossing London by Uber, as there are plenty of excellent public transport options.