Mt. Wachusett is a fun place to spend some time. Start out at the Visitors center on Mountain Road just south (above) the ski area. Pick up a trail map. From there you can go to trails on any side of the mountain.
The most popular (summer) trail is Pine Hill, leaving from the visitors center. It is an old ski trail, with much recent rock work, and it climbs steeply. It is icy in the winter. You can also go further south along Bicentennial Trail, connecting to Loop Trail, Mountain House Trail, or further to High Meadow Trail, one of my favorites.
Parking further south on Mountain Road, you can hike up Echo Lake Road to High Meadow Trail. If you drive down Mountain Road to Westminister Road, parking at the Administration Road gate, you can hike on Echo Lake Trail through a nice pine forest, or Stage Coach Trail past the windmill farm. You can loop back by many options in this area, including the Jack Frost or Harrington Trails.
Driving further out Westminister Road you can park at West Side Trail, and loop back by West Road or North Road. Any of these unpaved roads are nice for hiking, and usually see less traffic than the other trails.
Right from the ski area parking lot, you can head up to Balanced Rock. From there, beware the Old Indian Trail is closed where it crosses the ski slopes. They'd rather not have hikers on the slopes.
That said, if you bring a trash bag to sit on, you can slide down the right side of the slopes from the summit to the lower road. That's a whole new experience, but don't let your son see you, as he'd probably be WAY embarressed! I personally live to embarress my teenagers!
All the trails going to the summit have steep stretches which will be icy and tricky without traction. You can loop around and spend several pleasant hours on this fun mountain.