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.
Lots of routes exist to the top of this peak although they are almost all moderate hikes and not technical.
Classic hikes include:
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.
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.
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
From the West
From the North
The Observation Deck
From North Conway either take 16 N or 302 N park at the Cog or Pinkham Notch.
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Mountain Project's determination of some of the classic, most popular, highest rated routes for Mount Washington:
Featured Route For Mount Washington
Glen Ellis Falls WI2 PG13 NH : *NH Ice and Alpine Climbing : ... : Pinkham Notch
Fun little climb. Crazy mix of water ice, cauliflower ice, and snow. You get a little bit of everything. Stay either far right or left, there is a lot of water running down the center. Once on top of the falls traverse left to where the path is and anchor to some trees, watch out for running water! It would be fairly easy to fall in, and the closer that you get to the center of the falls, the thinner the ice gets. Not the time to go for a swim....[more] Browse More Classics in NH