I'm not a native London-ite but from my experience, you could either take the Tube (inexpensive but you lug your bags) or hire a car (no trouble with bags, but costs a bit).
Easiest route from Heathrow is Piccadilly to Earl's Court or South Kensington, where you can get on circle or district eastbound to tower hill, by butler's wharf. I know there is some construction going on at the moment, and I thought maybe it was in South Kensington, though it depends when you're riding.
The Heathrow Express seems like a waste, b/c you still have to get from pad station to Tower Hill.
There is a map and other information online somewhere, just google london underground.