North Cascades National Park: Drive through Marblemount to the end of the Cascade River Road and take the trail to Cascade Pass (about 3.5 miles). For better views, take the trail from Cascade Pass north up Sahale Arm. This will open up probably some of the most expansive views in the North Cascades. This is a very popular and easy trail.
Olympic National Park: Drive up to Hurricane Ridge and hike any of the numerous trails up there. The Olympic Mountains have the disadvantage (advantage?) that most of the alpine terrain is attainable only through very long approaches and multi-day backpacks. Hurricane Ridge is one of the few exceptions.