You can take Gatwick Express train service, It runs non-stop every 15 minutes with a journey time of 30 minutes.
The answer to this question (and many others) will change in late 2018 or early 2019 when the Elizabeth Line 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 an Elizabeth Line 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 Elizabeth Line 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!
I realise that this an older question, but I’ve done this journey many times, and the easiest way in my opinion is to take the Thameslink service, as stated in the other answers, but to remain on until Farringdon. Farringdon is a much easier change when you have luggage as the railway lines run in parallel with the underground. It also has a shorter journey to Paddington (about 14 minutes on either Circle Line or Hammersmith & City Line).
You can purchase a combined rail and tube ticket at Gatwick, but trying to find it in the automated ticket machine can be tricky, so I would suggest going to the manned ticket office, ask for a ticket to London including a single tube journey to Paddington, travelling with Thameslink (was FCC/First Capital Connect until late 2014). The ticket will read something like Gatwick Airport to London U1 (U1 being the underground zone that Paddington is in). The train you want will probably have a have a destination of Bedford, or Luton. But the signboards at Gatwick will tell you the destination, calling points, and operating company.
I would suggest the cheapest route would be to get a Thamelink train to London Bridge (not Gatwick Express, this is an unnecessary expense in my opinion) this takes 28 mins. Then get the Westbound Jubilee Line (Grey Line) to Baker Street. Cross the platform to the Bakerloo line (quick interchange) Northbound 3 stops to Paddington.
The most straightforward way would be taking the Gatwick Express to Victoria Station (the journey is about 30 mins). The Gatwick Station is directly connected to the South Terminal. From the North Terminal it’s a bit further and you need to take the automated shuttle train.
Trains leave every 15 minutes.
From Victoria Station you have no less than three options:
For all those options, make sure you use the exit at the front of the train, not the exit on the upper floor through the elevators or escalators.
You could also take a taxi, if the price isn’t a concern for you.
There are other possibilities I left out because they won’t be faster or more straightforward than this.
Note: be aware of the fact that, from time to time, especially during weekends/holidays, Gatwick Express has to make alterations to train services, usually for planned Improvement works.
Credit:stackoverflow.com‘