Mount Washington is a great mountain, and much can be said about it. I guess I'll begin, but all of your experiences will be helpful to fill out the possibilities of this unique mountain.
Mountain Washington is 6,288 ft tall and rises 3000+ft from the surrounding valleys and makes for an obvious landmark and it is the highest point around for hundreds of miles in any direction. For this reason, as soon as northern New Hampshire was starting to be explored people thought that Mount Washington was something special and therefore wanted to be at the top of it...
Many routes exist to the top of this formidable an fickle peak.
Comes from East, Drive or be driven to the Summit(Summer only)
Comes from the west, Expensive and antiquated but always a favorite of tourons (beware if hiking along the black smoke that comes out of it is really toxic, use something to cover your mouth)
Hiking: Lots of routes exist to the top of this peak although they are almost all moderate hikes and not technical.
Classic hikes include:
The Presidential Traverse (North to South 27 miles +/-).
The Amanoosuc Ravine Trail (shortest but steepest trail to Lake of the Clouds then contiue North up the summit dome).
Tuckerman/Huntinton Ravine Trails (common wisdom says don't try to hike back down these trails, they are too steep). Many other quality hikes exist but they are to numerous to name here, Check out the White Mountain trail maps/guides.
Glen Boulder Trail (bring a strong spotter and a pair of shoes, and your courage and enjoy some great lines on this boulder with serious exposure about 2mi up the trail.
Skiing: Tuckerman Ravine is the best extreme skiing on the East Coast! Be ready for crowds, over 2000 people can be there on any one day during the season (March-April). Beware of Crevasses that open up when Thousand Streams breaks through the snow.
Technical Routes: Many Ice routes exist up Huntington Ravine , but I have no personal experience, HELP! Many Rock routes exist and even more bouldering potential exists but it would have to be done so not to trample the delicate moss and apline garden that only exists up in the meadows around the summit.
Weather: Mount Washington is home to the worst recorded weather on the planet, and averages only 100 days of sun a year. Three different weather system converge at Mount Washington that make the weather here very hard to predict (no it isn't your weatherman) and very dangerous. Make sure to come way over prepared and ready to bail at any moment! Lake of the Clouds hut has an emergency shelter if you get caught up top. The Summit Weather Station is manned year round so if you really need help, you can always try there.
Check out the book "Not Without Peril" it tells stories of all different people over the past 100+ years and how they have had adventures, disappeared or met their maker on this formidable peak.
Also Check out these live images of Mt. Washington