According to South Somerset.gov website there are over 9000 public rights of way in the county totalling 3808 miles (6129 km). There are long distance footpaths through the surrounding area such as the Leland Trail, the Liberty Trail and the River Parrett Trail and the area around Montacute has lots of footpaths and green lanes (see screenshot below from footpathmap.co.uk). In the immediate area around Montacute the National Trust has some great walks signposted. Below I've included my favourite circular routes that can be taken around Montacute; the National Trust footpath routes can change to minimise erosion, if in doubt follow their signposted tracks. Please keep to the proper footpaths with dogs on leads where directed and please pick up after them; where dogs run around, so do children!
The screenshot below (from footpathmap.co.uk) shows footpaths in red and permissive paths in dotted red lines
The route shown below (marked up in red and screenshotted from Google Earth) shows a 1 mile walk that can be taken starting in the Borough opposite the Phelips Arms and the entrance to Montacute House, and going north through the village (this shows the route through Wash Lane but it is just as nice to walk up to the Kings Arms Inn (http://thekingsarmsinn.co.uk/) and St Catherine's Church then north along Bishopston). It passes the west of Montacute House with an iconic view down the drive, then goes down to the bottom of the park, back up around the formal gardens then along the east aspect of the house with its long avenue of trees through the park, and around into back lane and then back past the Toy Museum to the Borough. The National Trust currently has this route waymarked in the opposite direction through Montacute Park as part of their 'St Michael's Hill' walk; follow their route where possible.
A slightly longer walk of 1 and a 1/4 miles can be taken from the Borough going south-east through the village and then up through Ladies Walk. At the right time of year the ground is carpeted with wood anemones and bluebells. At the top of the walk there is a fantastic view of the Abbey Farmhouse (the old Priory gatehouse) and the village. The path is signposted by the National trust and comes out from the woods into a field, from here go left to exit onto Hollow Lane. Almost immediately opposite the field gate is a farm track, this leads you down past Abbey Farmhouse, past the old primary school and the church, then right down Middle Street towards the entrance of Montacute House and the Borough.
A longer circular route of about 4 miles takes in both St Michael's Hill with its prominent Tower and also Ham Hill, where the local stone is quarried. Again starting and finishing at the Borough, the nicest route to take is up through the village towards the church, left to the Abbey Farmhouse, then follow the footpath signs to go in front of the farmhouse towards St Michael's Hill. Skirt the south of the Hill then cross to the south-west corner of the field towards the woods ahead of you, there is a stile into the route across the top of Hedgecock Hill. Follow this route all the way to Ham Hill Country park. There's a fantastic view to the west from the lay-by where the ice-cream van often parks. Continue to the north of the Hill to admire the view from the War Memorial. From here walk clockwise around the north spur of the Hill, past the Prince of Wales and back down to the lower route home through the woods. When you come out of the woods follow the lane round to come into the field below St Michael's Hill. Head towards the gate and continue up to the Tower. Once you have admired the views from the Tower you can head back down the same route or due south down the steeper path, then back though the village.
This next route is also about 4 miles and starts off in the same direction, across Hedgecock Hill, but then follows the route down from the Hill to the picturesque Mill in Little Norton. From here walk east along the road past Tinkers Bubble until you come to a field gate on your left. Walk up though the valley; this valley was the site of the Medieval village of Witcombe, a village which was deserted in the 1600's. From the top of the valley walk right along the road then left down Hollow Lane; this is worth a few photos! The route marked on the Google Earth screenshot below takes you back through Townsend and past the Post Office and Toy Museum in South Street to return to the Borough.