Hi Sylvia
I do the Gatwick/ Heathrow hop quite regularly as you can't fly directly to Heathrow from Jersey (where I live). It's a very reliable service.
National Express is very regular and, depending on the time of day you are travelling and allowing for hold ups on the M25, usually takes about an hour and 15 (also depending on which terminals you are picking up and dropping off at).
It's easiest to book online (and cheaper) than at the airport itself and you can get a changeable ticket for no extra cost in case your flight is early/late and you need to travel at a different time. The number of the bus you have booked will be printed on your ticket and will also appear on the information board listing the stand/bus stop number it will be parked, so you don't need to worry about getting on the wrong one. The destination listed on the front of the bus probably won't say Gatwick or Heathrow because they usually go on elsewhere, such as Birmingham or the South of England, rather than just shuttling between the airports.
Yes, the drivers will load and unload your luggage for you and are usually pretty friendly and helpful. (Unless there's a jam on the M25!)
All the best
Mandy