The answer to this question (and many others) will change in late 2018 or early 2019 when Crossrail opens.
For future reference, board a Thameslink train at Gatwick (the likely destinations for this service are Bedford, Luton, Peterborough, Cambridge or King's Lynn) and change at Farringdon for a Crossrail train to Paddington.
Because this journey will be entirely on National Rail, you can get a through ticket without needing Underground - a normal ticket to "London Terminals" should be valid on the entire route.
Note that if you are travelling to Paddington for Heathrow, then the Crossrail service from Farringdon will include direct trains to Heathrow Airport (the main station at Terminals 1-3; you may need to change there for a shuttle train to T4 or T5).
Hopefully it will be a bit less than the £66.50 that Gatwick Express + Heathrow Express currently costs!