I curated this list of hikes to maximize great alpine scenery, quality of trails and minimize drive time from Seattle. All hikes are around 11-12 miles round trip, although all can be extended and many side attractions exist. All trailheads are within a 3 hour drive of Seattle, traffic dependent.
Colchuck Lake in the Central Cascades/Enchantments, accessed from Leavenworth, Washington. A fine 5.5 mile hike ends at the deep moraine Colchuck Lake. You will be treated to a 270 degree vista of 3,000 foot high granite walls, ridges, and couloirs of Colchuck Balanced Rock, Dragontail Peak, and Colchuck Peak. A photographer/climber friend believes it to be the most beautiful spot in the Cascades.
Mt Baker, Ptarmigan Ridge. Starting from the end of the Mount Baker Hwy, Artist's Point, you will enjoy the classic view of one of the most photographed mountains in the world, Mt. Shuksan. Look to the other direction and you will be looking at the steep Park Glacier headwall of Mt. Baker. The Ptarmigan Ridge trail hikes towards Mt Baker along a ridge, passing lush meadows, cascading waterfalls, and deep glacier carved valleys. In about 5.5 miles the trail abruptly ends at a knoll perfect for lunch and viewing Mt. Baker.
Cascade Pass. Already mentioned but worth a second vote. Hike the easily graded trail to Cascade Pass and enjoy views into ultra classic mountaineering objectives like Torment Peak, Forbidden Peak, Sahale Peak, Mixup Peak, and the mile high 'Eiger of the Cascades', Johannesberg Peak.
Ingall's Lake. This lake is accessed from I-90, just past the town of Cle Elum. The lake is gained after about 4 miles of enjoyable trail through a landscape dominated by olivine/dunite rock formations and mountains oxidized to a deep red. From Ingall's Lake look across the deep Ingall's Creek valley to the immense bulk of the perfect pyramid of Mt. Stuart, one of the biggest chunks of granite in North America. The creek valley neatly dissects the two geological areas; olivine to the south, white grandiorite to the north.
If you feel compelled to hike Mt Rainier, I recommend Spray Park from Mowich Lake. This, the most accessible trailhead to Seattle, boasts amazing wildflower meadows and views into the 'steep' side of Mt Rainier including Ptarmigan Ridge and Mowich Face. It is also less visited than Paradise, Sunrise, or White River so you will enjoy a more wilderness experience. The hike is about 5 miles, but one can hike farther up Ptarmigan Ridge or explore side attractions like Observation Rock.