If you are price conscious then London Underground Picaddilly Line the cheapest from LHR.
Have a look a the fares here:
http://www.tfl.gov.uk/tickets/14416.aspx
You can buy an oyster card from any station. If you are travelling at peak times ( 06:30-09:30 and from 16.00-19-00 Monday to Friday ) or plan to take lots of trips each day on buses and underground then the 7 day travelcard is the best option. Otherwise for off-peak trips into and out of London you could save money by using the pay as you go oyster card. LHR is in Zone 6 and the city centre is Zone 1.
Oyster cards are not valid on Heathrow Express or Connect from LHR, and while it is faster it is also much more expensive.