If the ski resorts aren't your thing, there's plenty of sno-parks offering something for snowshoers, cross-country skiers, and
This article from the Washington Trails Association is a great place to start for tons of information on snowshoeing. For details on snowshoeing and other winter recreation on DNR-managed land, see this post on their blog.
Plenty of excellent snowshoeing and cross-country skiing can be found at Washington's Sno-Parks. There are also motorized Sno-Parks for snowmobiles, and Sno-Park play areas.
The closest Sno-Parks to Seattle are along or near I-90 and include:
There are several along Highway 2, a few miles outside of Leavenworth:
And of course there are more all over the state, see the Washington State Park's site for a complete list and maps.
For information on the passes required, see this pdf from the Washington State DNR.
Other Regional Winter Recreation Areas
Always remember to check the latest avalanche conditions before starting out!
Several rewarding snowshoe routes start right behind the Paradise Visitor Center.
The Panorama Point route is a strenuous 5-mile route with more than a thousand feet of climbing and amazing views. It follows the Skyline Trail to Panorama Point and then down along Edith Creek.
The nearby Mount Tahoma Trail System near Mt Rainier has over 50 miles of winter trails and a system of awesome huts available for overnight stays.
Mt St Helens
Mt St Helen's offers numerous snowshoe and cross-country skiing opportunities. For the adventurous, you can climb via the winter route, or explore the other offerings at the Marble Mountain Sno-Park:
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This Forest Service map should have all the info you need. Click on the thumbnail to see the full version. A sno-pass is required.
The Echo Ridge Trail System near Lake Chelan is another option for cross-country skiing and snowshoeing. A map from the Forest Service is available here, and a vicinity map and detailed conditions are available at lakechelannordic.org.
On the Olympic Peninsula, Hurricane Ridge is another good option, there are a variety of winter activities. There are several miles of winter trails and downhill skiing. The are also guided snowshoe walks during parts of the season. Always check the latest conditions.
Mt Baker and vicinity has several nice snowshoe trails and varying lengths and difficultly. These and other routes can be found in Dan A Nelson's book, Snowshoe Routes Washington. The WTA has licensed the book, and all routes are available online as descriptions of the individual routes.
The following trails can all be found along 542 beyond the town of Glacier and the Mt Baker Ski Area
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Starting at the eastern end of 542, the Mt Baker Ski Area is the starting point for several hikes of varying difficulty. The centerpiece is the 5-6 mile one-way snowshoe to Artist's Point, but there is a wide area to explore, including Austin Pass and Kulshan Ridge. For an easier snowshoe, try the Bagley Lakes Trail which starts from the same area.