Train, as long as it is running (check in advance!) is great. Gatwick to East Croydon, hop platform, Caterham train to Forest Hill, done. We travelled into Gatwick that way about a month ago on a weekday morning, and marvelled that we'd walked no more than 5 minutes between front door to check-in

If that option isn't available, the changes can be horrific (as they were on our weekend journey home) and you'll wish you'd got a cab!