start at Pinkham Notch as for all of Huntington Ravine. When you get to the bottom of the ravine, the best approach in my opinion is not directly up the gully but rather up another shallow gully which leads to sort of a dead end in the rocky wall between South and Odells Gully (South being the one farther left). Once you are close to this wall, climb a short slope which takes you out to a wide spot which is the entrance to the real "gully" part of South Gully.
You're now close to the entrance to the classic rock-walled couloir that the first part of South Gully is. Here on your right is yet another shallow gully which is not obvious to everyone and not named, but is actually not a bad alternative snow climb (no ice here at all) all the way up to the Alpine Garden - a safer alternative (though fairly steep) to South Gully when there is avalanche danger. For South Gully go straight up; the gully soon becomes deep and increasingly steep, and at one point it takes a sharp left turn. Here is an interesting section, because on one hand it's straightforward snow no more than 45 degrees, but on the other hand if you don't self-arrest immediately, you go straight into a rock wall because of the corner. When a friend and I did the gully we climbed this part unroped, but it may be wise to use a running belay with pickets if conditions permit. A couple hundred feet up is the crux of the route - a VERY SHORT and i mean REALLY REALLY short bulge of ice. BUT: don't understimate this section because of how short it is. The reason i make a special warning here is that it's really easy to cruise, unroped and without any of your gear ready, up the snow right up to the ice, and then find yourself in a jam, stuck on steep ice with only a mountaineering axe and no protection at all, with a potentially deadly runout below you. DONT LET THIS HAPPEN and get ready well before the ice bulge; if you do you should have no problem - generally a simple anchor of an ice axe and a picket should suffice for the whole thing. If you're really conservative take ONE screw, that should definitely be enough. Incidentally, for those who like rock, you can bypass the ice altogether by a short easy rock climb on the left - if you dont mind giving your crampons a beating of course...
The remainder of the gully is straightforward snow climbing; we did it unroped, but again depending on your partys experience, snow conditions and your overall feeling you could do a running belay here with pickets. The gully exits on the Alpine Garden between Odells and the Escape Hatch; the Escape Hatch is a very convenient descent and just a few hundred feet left. Some parties go back down South Gully, rappelling or downclimbing the ice bit.
ice axe (possibly a second tool, but can be done with 1), crampons, rope, pickets (at least 2), possibly 1 or 2 ice screws.
If you have information about this route that doesn't pertain to any of the other sections, please add it here.