Upriver of Kingston bridge it's Ravens Ait which used to be sea cadets, downriver, there's Ariel (BBC Sailing) on the Middlesex side and on the Surrey side there's Albany Park - which is a council run youth club for sailing and canoeing next door to a Sea Scout group (possibly Ajax).
Hampton Sailing Club is on the web at (spookily enough) www.hamptonsailingclub.com , where you will find all the details on how to join. It's a very friendly family-oriented club, which still has members from when it was set up 60 years ago, and you can often see 3 generations of same family on water. I spent 20 happy years as a member there, on one of the nicest stretches of the Thames for sailing. They race every Sunday from March until Boxing Day, with boats ranging from Enterprises and GP14s (suitable for beginners), Merlin Rockets and Lasers for the more athletic.