There are a lot of great "walks" in the Kingdom. Two of the most popular and accessible trails are the hike up Mount Pisgah overlooking Lake Willoughby and the hike up Bear Mountain overlooking Lake Memphremagog (with views of Jay Peak to the west). Great hikes for all ages and wonderful spots for picnic lunches at the top. You can get a wealth of information and maps on the website of the Green Mountain Club, Northeast Kingdom Division by clicking here.
And speaking of great spots for picnics, Big Falls in North Troy is one of the best kept secrets of the Kingdom. A great swimming hole and a wonderful place to just hang out for hours or for days. Bring a book or two and let the sound of the falls bring some peace and "quiet." You can drive right in, so this isn't really a "hike" unless you like to scramble around the rocks a bit. Head out from Newport on Route 100 heading West (100S) past East Hill Road is the River Road. Take that heading north and you'll find it.
We never get tired of these day trips. But if you are hungy for more, check out the GMC NEK Division web site.