We are looking to get to Canterbury with lots of luggage without a lot of stops from Heathrow airport.
Any ideas and prices?

First, I'm not going to hector you about your luggage. I'm assuming you're carrying whatever bags you have for good reason.

IMO, the path of least resistance would be to book a car service to take you directly from LHR to Charing Cross or Victoria stations for a train to Canterbury (does it matter whether you go to Canterbury East or West?). Car service would be about £30 to £40, more or less and there are many recs for car services here on Fodors. You can check train fares at www.nationalrail.co.uk

However, other options include Heathrow Express to Paddington, cab to Victoria or Charing Cross, train to Canterbury (which will probably cost you more than a car service direct from LHR to your train station)

Or the Tube from LHR to your train station, but I believe that requires at least one Tube line change. Cheapest, but least appealing option, IMO.

Though this site is how to get to Canterbury University, on the edge of Canterbury, check it out for Heathrow-Canterbury transporation

How to get to/from Canterbury Hall from/to Gatwick and Heathrow ...
How to get to/from Canterbury Hall from/to Gatwick and Heathrow Airports ... Getting there by bus. Gatwick Airport is well connected by National Express ...
http://www.lse.ac.uk/accommodation/U...ting_there.htm

SCARTCH the above - mixed up Canterbury Hall in London University - is not in Canterbury

Sorry for faux pas

The discount fare for paying cash using www.justairports.com (a car service recommended by many Fodorites) from LHR to Victoria is £49 for a saloon (sedan) and £53 for an estate (station wagon). Since you'll have a lot of luggage, you'll need the estate.

If you're able--and it sounds like you may not be--to schlepp your own luggage from Victoria coach station to Victoria train station (about a 15-minute walk), you can take a national Express coach directly from LHR to Victoria. The luggage goes in the hold and the fare is £4.

Oops! I clicked on Gatwick rather than Heathrow on the Just Airports site. The cash fare for a saloon is £30 and for an estate it is £34.

You could also take a National Express coach to Victoria and then another to Canterbury, but you'd have at least a 40 minute wait between coaches.

The fare for the route suggested by thursdaysd is only £16. And your luggage schlepping would be minimal.