In our opinion, the best way to enjoy the White Mountains of New Hampshire is to get outside. Depending on your skill and fitness level, there are options ranging from paved trails that provide access to beautiful waterfalls to rigorous hikes with significant elevation gains ending on mountain peaks.
Keep in mind that there is a parking fee at the majority of the White Mountain National Forest trailhead parking lots. You can pay per day at the self service pay center at each trailhead, or stop in at one of the many ranger stations for weekly or annual parking passes. See the National Forest website for more details.
We highly recommend the Appalachian Mountain Club's White Mountain Guide for planning hiking trips. It documents nearly every trail in the area with mileage, elevation gains, and general descriptions; with so many trails out there it's essential for anyone doing serious hiking. Plus, the maps in the newer versions are great. See the AMC website to buy the guide online.
The White Mountain Region offers a plethora of cross country skiing options, from ski touring centers to back country adventures; depending on how good the season is and how much snow has fallen, you can get up to four months of good skiing in the area.
For touring centers, we particularly enjoy Bretton Woods, located right at the base of Mount Washington, and Waterville Valley, which is a little further south. Both have fantastic trails and generally tend to get pretty good snow. The Bretton Woods location has trails right around the Mount Washington hotel. You can rent skis at both locations, and they do a good job of updating their websites with information about open trails, while they'll also give helpful information on the day of your visit. If you have the time to head further north in New Hampshire, the Balsams resort also has some spectacular skiing, and they also have great food to top off the day on the trails!