I know a few places
Kinda depends what you like and how far you want to walk
- Horner Woods : large ancient oak woodland
- Coleridge Way : He allegedly walked it in a day
- Mineral line : interesting local history
- Dunkery Beacon : county top and open moorland
- Pinkery Ponds up to the Chains : proper scary moorland
- Doone Valley : of Lorna Doone noteriety
- Two moors way
- Tarka trail
- Macmillan Way
- South West Coast Path
Plus you can walk out from Brendon two gates to find a lost village or even from Timerscombe to find the lost village of Clicket
My fave half day stroll is Larkbarrow out to Three combes foot
Many many more obviously, but those spring to mind first !
Part of the pleasure is in the planning so I wont give spoilers
Have fun