The train system is excellent and very moderate in cost. Your local bookstore should have maps and schedules. Purchasing round trip tickets is a savings too. We went directly from Heathrow to Victoria Station and then took a train directly to Dover.
It looks like there are no elevators (lifts) in the train stations but there really are, they're just hidden. You'll need the lifts with your luggage.
In Dover take the double decker bus tour and visit Dover Castle if you have time. It's fun to see and has lovely views of the area.