The most straightforward way would be taking the Gatwick Express to Victoria Station (the journey is about 30 mins). The Gatwick Station is a 5 mins walk from the South Terminal. From the North Terminal it's a bit further and you need to take a bus.
Trains leave every 15 minutes.
From Victoria Station you have no less than three options:
- take the underground/tube/metro/whatever-you-call-it Circle Line
(High Street Kensington direction) and get out at Paddington stop.
It's gonna take about 20 minutes.
- Accordingly to Gagravarr's comment you could also go from Victoria Line to Oxford Circus, then a cross-platform change to the Bakerloo Line up to
- If the metro is closed, take the bus 436 to Paddington.
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.