These pretty woods near the village of High Littleton have some nice woodland trails to try. There's free car parking and in the spring there's lots of bluebells within the ancient, mixed woodland. There's also a lovely waterfall and the option of picking up the Limestone Link which skirts the southern edge of the woods. You could pick up this long distance trail and head towards Clutton, Temple Cloud or Timsbury. In nearby Clutton you can join with the Three Peaks Walk and further explore the beautiful Chew Valley. Heading west will take you to the splendid Chew Valley Lake.
Greyfield Woods are located not far from both Bristol and Bath.