Road riding is pretty good in that area, you can get out to the Cotswolds easily enough and out into Worcestershire in the opposite direction. There are lots of pretty villages and country lanes in both directions.
Off road is more of a challenge. Lickey and Clent are 20 mins drive away, both are good fun for a few hours but neither are a huge area. The Wyre Forest is maybe 40 mins away and is good though you need to befriend a local to get the best out of it. Kinver is a small area and quite steep but, again with local assistance, can be good fun.
Edit, just seen the Sambourne comment, stay on the road, loads of really nice road around there, and lots of good cricket clubs to stop at on the ride and have a pint as well.