Cheltenham is very close to the place I was born, the other side of Gloucester in a little farming village on the River Severn.
There are many places to visit from there. Gaydon has been mentioned, not far from there is Stratford on Avon, Oxford is within easy strking distance and there are innumerable small towns and villages in the Cotswold hills. You could visit Northleach, just off the A40 between Cheltenham and Oxford.
If you're interested in birds, there's the Severn Wildfowl trust at Slimbridge, going south, or go further south to Bath, from where Stonehenge is easily accessible.
Or go west, towards Ross-on-Wye, and drive down the Wye valley to Chepstow, passing through Monmouth and the ruins of Tintern Abbey. Then you could come back across the Severn bridge.
Though I am naturally biased, it really is lovely countryside, and I get very nostalgic thinking about it!
The photo is of a pub on the river close to my birthplace. I was there back in the summer for lunch.
Anchor Inn, Epney, Gloucester.