We are staying in the Bayswater area north of Hyde Park. What would be the best/ cheapest way to get from there to Gatwick airport?
Underground (Circle Line) from Bayswater to Victoria station, then the Gatwick Express from Victoria to the airport. Should take about an hour in total. Journey planner here: http://www.tfl.gov.uk/
You'll save money if you take a Southern train rather than the Gatwick Express from Victoria to Gatwick. Southern: Anytime for £14.40, off-peak for £12.80, as little as £5.00 if bought online well in advance. Note, however, that a discount fare ticket is only good for a specific departure date and time and is non-returnable and non-exchangeable. Gatwick Express: £19.90 if bought at the station and £17.75 if bought online.