I'm afraid that you're going to have to become intimately involved with public trans. There may be other ways to see London (and the rest of Europe) but there are none cheaper nor easier nor faster.

Depending on your airline/flight you'll probably arrive at LHR. And Depending on where you're staying will depend on which line you take. But most are cross-connected anyway. If you are staying around London Paddington then take the Heathrow Express. The other option is the Piccadilly Line that makes numerous stops. You'll probably arrive in the morning so the train will be crowded.

First determine what hotel you're at and then find out what tube stop is closest. Then pull up a map on the internet to determine your route. There are tube maps available in nearly every guidebook and are usually available at most stations.

Your first trip on the tube may be a little daunting but once you use it you'll fall in love with subways and could never think of using anything else no matter what city you visit. Just study your map and follow the signs. Also, think of it as an adventure. If you get lost, so what? Just backtrack and start over. It'll make you all the more familiar.