285 Bus from Heathrow (Central Bus Station) to Kingston .. get off at Feltham Station and hop on the train - there's frequent trains from there to Twickenham (takes about 5 minutes, just be careful as not all trains towards London from Feltham stop at Twickenham)
Get yourself an Oyster Card if you haven't got one - I dont' think you can buy them in the Bus Station at Heathrow but you can get them in the Tube Station downstairs .. it's much cheaper to make journeys with Oyster compared to cash fares ... Get an Oyster Card and put a tenner on it will easily get you to Twickenham and back.