If you can afford it, get a minicab. It will probably cost you about £20-£25 which is a lot chreaper than a black cab and about the same as if you got the tube + Heathrow Express. Just make sure you agree the price beforehand - most minicab firms have a set rate to Heathrow. If you want a minicab to collect you when you come back, you can arrange that in advance too (since minicabs are not allowed to tout for fares at the airport) - you give them your flight details and they send a driver to meet you. You could also get a cab or tube to Paddington and Heathrow Express from there, but like I said I don't think it will save you much money. I am not sure if you can still check in at Paddington with some airlines, that stopped for a while, your airline can advise you on that - it means you get to dtich your bags there and get spend time hanging out in Paddington (ooooh how exciting! NOT) or head straight for the excellent duty free shops at Heathrow. I LOVE the airside shops at Terminal 4!