City Airport is central and lovely but not relevant to you. Because the runway is part built over the river it is only a short runway, and therefore only has the capacity for smaller planes. Smaller planes take less fuel, ergo transatlantic flights cannot go there. I suppose if you had a connection from Ireland then you might find yourself on a flight into City, but I doubt it.
I am probably going to repeat what others have said, but this is my feeling on the matter:
Heathrow is easy to get out of on the Heathrow Express, two normal mainline routes (one to Paddington, one to Waterloo) and the expensive Heathrow Express. There is also the tube, but may want to consider the time you are landing if thinking of going this route, as getting the tube during rush hour after a long haul flight with big bags is NOT fun.
Gatwick is, as others have said, a train, and I would say that it is not really worth paying the extra for the Gatwick Express - it doesn't really shave much off your travel time.
It might help if I knew where in London you are staying (sorry if you have already said) as this might make a difference.
I am assuming that by 'trolley' you mean 'tram'....the best form of transport known to man!!!! I LOVE trams!!!
Vicky (transport geek)