Hi there TommyTD... and welcome to Banbury. Home of the delicious coffee smells every week or so.
The fisheries I know within reasonable striking distance:
Farmoor - just west of Oxford. Not the most attractive of venues but highly rated by all who fish there
Draycote Reservoir nr Southam. Highly rated (esp by my neighbour!) large resevoir
http://www.lyttlesofdunchurch.co.uk/draycote.html
Salford Trout Lakes - just beyond Chipping Norton. Two small to medium size stillwaters in very attractive surroundings
http://www.salfordtroutlakes.co.uk/
Heathcote Lakes - not far up the M40 and has a decent reputation
http://www.heathcotelake.co.uk
Packington - a little further up the M40 near Meriden - lots of good feedback
http://www.packingtonestate.net/fisheries_trout.html
Lenches Lakes nr Evesham. Similar to Salford but a little smaller. URL="http://www.lencheslakes.co.uk"]http://www.lencheslakes.co.uk[/URL]
To the West is
Bransford Fishery (in the vicinity of Worcester/Kidderminster, around 140 miles round trip)
http://www.bransfordgame.com/
As per another post to the East and North within 2 hours max travelling are the Leicestershire/Northants/Rutland Fisheries - Eyebrook, Elinor, Pitsford, Rutland, Ravensthorpe etc....
To the south west Bushyleaze and Lechlade are not too far.
As for river fishing... a little farther afield I'm afraid. There is some fishing on the Colne and Windrush, although not easy to get access. To the West there are some Wye & Usk Foundation waters on Rivers Wye (not surprisingly), Arrow & Lugg around Leominster.
There is an Orvis shop in Burford that has some details to and are invariably helpful. I have it on good authority that Lyttles of Dunchurch are very good too.
The two tackle shops in Banbury are predominantly Coarse based and I have never asked any Fly related questions. If you do I would suggest Banbury Gunsmiths. They do have a very small amount of fly tackle at least (and I mean small).
Also worth a look is
http://www.troutfisheries.co.uk/central.htm
That should keep you going.
All the best,
Jon