Wells in Somerset or Wells-next-the-Sea?

There is a daily direct bus with National Express from London to Swaffham, which would be the easiest option. There's only a slight time saving by going by a train/taxi or train/local bus route, but this might be necessary depending what time of day you want to travel. The nearest station is Kings Lynn.
Swaffham to Norfolk - looks to be a pain, with the only bus option a trek back to and out again Kings Lynn. Transport Direct for buses. (Input 'Wells-next-the-sea' to find the Norfolk one....assuming that it the one you want!)

The cheapest option, and quickest, is the train from London King's Cross to King's Lynn. Then you can catch the coasthopper bus along the coast. The best place for times etc is http://www.travelineeastanglia.co.uk
The north Norfolk coast is superb in September, especially as there are not quite so many visitors. The beaches are beautiful, walking is excellent and you can really enjoy the pubs and restaurants.
A couple of useful sites for you are:
http://www.brancasterstaithe.co.uk
http://www.itnorfolk.co.uk
Have a great trip and maybe we'll see you there.