Partway up Mt. Adams
Mt. Madison is a mountain on the northern side of the Presidential Range. At 5,367 feet, it is the fifth-highest peak in the range. It is often hiked in conjunction with Mt. Adams.
Get online directions to Gorham, NH. The best place to start an assault on Madison is the Appalachia Trailhead, which is just west of Gorham on Rt 2, very close to Billings Cabin.
There are a number of routes up and down Madison - details are available in the White Mountain Guide. One popular route is to follow the Valley Way up to the Madison Spring hut, take a short hike up Madison, then take the Airline Trail back down (or the reverse).
The Valley Way follows (you guessed it!) a valley most of the way to the top, and passes some lovely cascades just after the trailhead (you may need to take side trails to see them all). This also means it is sheltered by trees almost all the way to the ridgeline, which it hits just before Madison Hut, where you can get water and shelter when it is open. From the hut, it's no more than half a mile to the summit of Madison.
To get to Mt. Adams, you have a few choices. The most straightforward is to take the Gulfside Trail a short distance over to the Airline Trail for the last scramble up to the summit. Instead, you could take the Star Lake Trail around behind the summit for some views of the other side of the ridge and interesting alpine lakes, adding just under a mile. In either case, the summit cone is bare, rocky, and steep. Views from the summit are stunning.
The Airline Trail parallels the Valley Way to the west, following the line of a shoulder ridge. It is much more exposed than Valley Way, providing some fantastic views, but little shelter from bad weather.
Advice and Anecdotes
Mt. Madison is a pretty dangerous hike in the winter - the summit is quite exposed, and its long snowfields present the opportunity for quite a long slide in the event of a fall. The Airline Trail has precipitous drops on both sides, and a fall in either direction could easily prove deadly. So if you want to hike Madison in the winter, ice axes and a little bit of self-arrest practice are strongly encouraged. CnT has had some close calls with uncontrolled slides here in the past, and it'd be best not to repeat them!
You'll also definitely want to bring crampons of some sort. Preferably, bring mountaineering boots and full-foot crampons from DOR (or hiking boots and full-foot strap-on crampons if any trippees have them). If need-be, in-step crampons could work, but they're a lot less reliable, especially on steep slopes.
Finally, you'll definitely want ski goggles and face protection for all your trippees - you'll be above treeline for quite a while, and Mt. Madison can have some very strong winds!