THANK you all for your advice! I truly appreciate it. Initally I was planning on finding a shuttle service and then read some comments on the express trains, but now have decided to go with a shuttle service.
I saw one recommendation for a shuttle service. Do any of you have recommendations for other reliable shuttles?
Again thanks everyone!

Regarding recomendations for car service, many people recommend Just Airports which certainly seems to be one of the least expensive, if you pay cash (they charge quite a bit to use a credit card). For years I've used Swiss Cottage Cars. They now have a website (for tourists) but seem to charge a lot more if you book on their website. I pick up the phone and call them at 011 44 207 431 2700 (to make it a really cheap call I first dial 101-6868 or 101-6400). I book a week or two in advance and get an exact quote when I give them the flight information and the address I'm going to.
There are others including Ray Skinner, whom everyone seems to love, but most charge quite a bit more -- some getting close to the cost of a taxi!

I'd recommend car service as well. We always have used Ray Skinner and Ray himself normally picks us up at Term 4. He gets a luggage trolley and wheels our stuff over to the "car park" where his car is always right handy to where we come in. It takes all of about 2 min. On the few occasions we have used other services it was exactly the same.

On going to Heathrow the drivers will drop you right in front of the terminal (just like you do at home) so you don't have to drag your luggage further then into the terminal.