You can go west on Route 30 where you will find a number of parks, swimming holes in the summer and hiking trails. Townshend State Park is about 25 minutes from downtown and has camping and hiking (moderate difficulty). Jamaica State Park can be reached by turning north in the center of Jamaica and crossing the West River. Camping and hiking here. The trail/road parallels the West River and is a beautiful, leisurely walk. Head north of town and you will find commercial camping sites at Brattleboro North KOA Campground and Cottages on Route 5. In Putney you can take Westminster West Road (west) and turn left onto West Hill Road. Then take Putney Mountain Road up to the crest of Putney Mountain. The short trail follows the crest of the ridge up to a wonderful overlook down into the West River valley.
If you want to head west on Route 9 (toward Bennington) you will find camping at the Molly Stark State Park. There is a nice trail (easy to moderate) up to a fire tower that allows you a sweeping view of the surrounding mountains. There are also some limited facilities and hiking in the Green Mountain National Forest. You will see several trail head parking lots along the western part of Route 9.
You can also take a short drive over into New Hampshire, east on Route 9 (off Exit 3). At about 5 miles, take a right onto Route 63 and go to the village of Chesterfield. There you will see signs for the trailhead of Pisgah State Park. There is a wonderful trail system that covers a variety of terrain. A little further east on Route 9 you will find the Chesterfield Gorge State Wayside on your left, a pleasant and easy hike in a picturesque gorge.
A word of caution: When you park your car to go hiking anywhere, be sure to lock your doors and don't leave valuables visible in the car. A little care will save a lot of trouble.