About 2-hour drive southeast of Seattle is Mount Rainier National Park. At 14,410' high, the mountain is actually quite visible from Seattle. It is the most glaciated peak in the lower 48 states and is an active volcano. Like the other Washington parks, it supports a variety of beautiful ecosystems - from old-growth forests to subalpine meadows to icy alpine tundras. The Paradise Jackson Visitor's Center is a great place to check out and includes a cafe.
For hiking on the east side of the park, I enjoyed the old growth trees on the 1-mile Grove of the Patriarchs Trail. There are numerous waterfalls in the area and Silver Falls Trail is a worthwhile 3-mile trip. On the west side, head to the The Comet Falls Trailhead. It's about four miles east of Longmire. The amazing Comet Falls Trail is a 3.8-mile round trip and Van Trump Park is 5.8 miles with about twice the elevation gain.
If you are looking for alpine gardens, consider the 5-mile Skyline Trail. It is just past the Visitor's Center. Note that even in the summer, I have dropped into a good amount of soft snow while hiking. So, if you wish to do a lot of hiking, be prepared. As a general rule, the southwest area tends to have a lot of trails under 5 miles, whereas the southeast, northwest, and northeast have mostly longer trails that will challenge experts.
For lodging, there are two options inside the park. The National Park Inn at Longmire is open year-round. When I visit in cold weather, I get an upstairs room and I'm usually able to put a drink and some coldcuts in the upstairs window sill for natural refrigeration. The Paradise Inn is a historic inn that has many more guest rooms and sits about a mile high. Both inns have a restaurant.