I'm a native of northern Massachusetts and will simply tell you that the White Dot and White Cross trails from the main ranger station are wicked busy and should be avoided. They are boring, no views. They are, in fact, the worst trails on the mountain; people take them because everyone else does, and when someone has been onm Monadnock once or twice, they do what the herd does.
The most direct route is the Old Toll House Road, which is 1.2 miles on a gravel road, then the old Half Way House area (with Moses Spring) and then it's a straight, rocky climb up the spiritless White Arrow trail, with some highly steep areas on the rocks near the summit. It's direct, but lacks much to see or enjoy.
Best trails to the summit:
Old Toll Road to Half Way House, then Monte Rosa trail to the weathervane, a quick detour to Amphitheater Trail to see Black Precipice (highest ledge on the Mountain), and the Smith Summit Trail to the top. I know, it sounds complicated, but it's not and the ranger in the booth at the base of Old Toll Road (the same one that collects your five bucks) will give you a map and directions. You will see the most "landmarks" by this route, and although overall it's maybe 0.5 miles longer than the direct White Arrow trail, you will have more privacy and views before hitting the summit. For instance, you'll see "The Tooth" and "The Billfold" and stuff like that. It's still a hard hike, but just more scenic.
Summary: Old Toll Trail to Monte Rosa Trail (sometimes called the Fairy Brook Trail) (just past Half Way House pasture, on the left) to Smith Summit Trail (with a tiny optional detour to Amphitheater/BlackPrecipice). This has the most features to see of all the hikes.
The best "for the strong hiker", but your ten-year old can do it, trail to the summit:
Birchtroft Trail (from Gilson Parking and Camping area) to Cascade Link (just before Red Spot Trail) to Spellman Trail (steepest climb on Monadnock, but oh, the views!) to the last 0.6 miles of Pumpelly Trail along the top ridge (the prettiest part of the Pumpelly Trail) to the summit. You can return from the summit on Red Spot Trail. Lovely views. Your best shot at seeing Boston. Spellman (0.7 miles) is very steep but clean rock. Do not go on Spellman after rain or in winter/ice...it's too steep, and has springs that run over the stones. You will fall. This is a great hike, I promise you. Any sturdy person can do it, but be ready for some challenge. I'm 50 years old, but in good shape, and I did it. You can, too. It's a real thrill, though. It's about 3.5 miles each way, which sounds long, but it's more gradual than the other trails. It is also on the opposite side of the Old Toll House-Monte Rosa Trail.
Summary: Birchtoft to Cascade Link to Spellman Trail to Pumpelly to summit. Return on Red Spot Trail.
Enjoy! I have hiked this mountain since I was a kid, it's the best. The best book about all these trails is Southern New Hampshire Trail Guide (Daniell and Smith). (plus it has a trail map). Enjoy! These trails sound hard, but each that I mention has signs on the trail. bye now!