Check out http://www.heathrowexpress.com/ Heathrow Express also stops at T1, 2, 3. Don't worry about tickets, buy them there.

Heathrow Express takes you to Paddington Station, which is not really central London. Unless you are staying in a hotel near Paddington Station, it will cost you still more money and the schlepping of your luggage to get to your ultimate destination. We (my wife, seven year old, and I) use the Underground to get into central London. Since the Picadilly line begins at Heathrow it's no trouble finding seats and we just stick our luggage somewhere close to us. Unfortunately, the Underground is not yet disabled-friendly, meaning there are places where you have to carry your luggage up stairs, but we've not found it all that bad, and it's a lot cheaper than the cost to Paddington and then cab to our destination.

Regards, Dave Tucson, AZ