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 First Capital Connect 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, traveling with first capital connect / thameslink (old name). 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.