Is there a direct way by train or coach, or do I have to go through London?

I have a National Rail map dated May 2001 which shows a route from Brighton to Lewes, Eastbourne, Hastings, Ashford, Canterbury. Requires at least one change.

www.thetrainline.com

You can't book tickets more than 3 months ahead and they won't mail tickets out of the UK

Brighton to Canterbury, shows a connection in Ashford - 2 1/2 hour journey

Thank you everybody! I will finally get to visit the Cathedral...

thetrainline requires yyou to have a sign-on ID before you can get prices
Try www.nationalrail.co.uk instead - and note the option when getting prices to select the "cheapest" as opposed to "fastest" route otherwise you'll always go through London

Thanks alanRow! That site worked just perfectly. I will be staying at the Univ. of Sussex, in Falmer, and I have now figured out how to go from there to Canterbury without passing through London.
My itinerary:
Falmer - Lewes
Lewes - Ashford International
Ashford - Canterbury West

All in 2:18 - marvelous!