I know you said you don't want to drive anywhere, but if you visit Stratford-upon-Avon, Warwick is only a short distance away. Warwick town is smaller, and less touristy (busy) but is still a nice place for a stroll. There is also Warwick Castle, which is well worth a visit.
At Stratford, you can rent boats and take a trip on the river. There are also some nearby National Trust properties - from memory where Shakespeare lived, and another where his wife grew up.