Hello!
I live in Leamington Spa with my British boyfriend. It is an amazing spot - I hope you like it. Actually, if you want help with finding a place to live, do get in touch - L.K.Wible@warwick.ac.uk. The way to look for places online is through newcomer link: http://www.rightmove.co.uk [nonactive] . You'll probably find that even if it looks kind of far away on the map, it's still probably quite close to the town centre but, of course, some parts are nicer than others. I don't know of any groups, but we'd love to take you for a pint.
If you like tennis, bring your rackets. Also, there's a great kayak club.
All the best,
Wible