Great as an introduction to hiking in the White Mountains, Mount Tecumseh is the shortest of the AMC's New Hampshire & New England 4000-footers at 4,003 feet.
This short hike starts out from the Tecumseh Trailhead parking lot along Tripoli Road at an elevation of 1,820 feet. The trail heads south from the parking area and almost immediately crosses a few brooks which may be difficult at high water. The first mile or so is a relatively gradual ascent.
Once you reach a cairn the trail makes a hard turn to the east and then begins to ascent to the first knob of the western ridge of the mountain. Once you reach the top of the first knob, there is a short side path (~100 ft) that has a nice view to the west. From this viewpoint you can see Moosilauke, another great White Mountains hike.
After this first knob the trail turns northeast for a short distance to the second knob then turns back to the southeast where you descent into a saddle before the final ascent to the main summit. The last push to the summit becomes pretty steep with a few scrambles. At the summit, there is a limited panoramic view to the east. Part of the view includes the ski slopes of the Waterville Valley Ski Area that uses the eastern side of the mountain.
To descend, just turn back around and head back. Just be careful on the initial descent going down the scrambles, they can be very tricky if wet.
Tripoli Road is a Forest Service road and is closed in the winter. While this would probably be a great winter hike, I am not sure how you would access the trailhead short of humping the road from either end. You can also hike Tecumseh from another trailhead that starts at the Waterville Valley Ski Area so that might be a winter option but I am not familiar with that side of the trail.