Catch the Heathrow Express to Paddington. It will cost £20 and there are ticket machines everywhere. If you don't have a ticket, you'll be able to buy one on the train (for a little extra).
Just get on the train if you arrive at Terminal 1, 3, or 5. If you come into T4, you need to catch a shuttle - just follow the instructions.
The trains leave every 15 minutes, and it's 15 minutes to Paddington from T1/3. Much better than an hour on the Piccadilly Line.
From Paddington, catch the Bakerloo line to Baker St, and change to the Jubilee to Southwark.
Bonus knowledge: Baker St is the oldest underground train station in the world, at some 160 years. Extra bonus: Southwark is pronounced "Suth-uck".
Buy your Oyster at Paddington. Get £20 credit. The Oyster ticket machines (and ticket desks) will only appear once you take the escalator underground: the on-the-surface machines are all for National Rail.