Rockwatch
The Club for Young Geologists. An award-winning kids-orientated club with colourful magazines, free fossils and unique expeditions. Win £100's worth of fossils in their yearly competition. Families join at a discount. Can't be recommended enough, made me what I am today! ;)

The Geological Association
Basically along the same lines as Rockwatch, but for professionals and adult enthusiasts. Founded in 1854.

Natural History Museum

Other Commonwealth

The Geological Society of Australia
Members receive good geological publications, discounted entry to conventions, discounted books on local geology etc.

The Geological Society of New Zealand
As above.

The Geological Association of Canada
As above.

United States of America

The National Museum of Natural History (Smithsonian)
As above