I have vacationed the last 4 summers in WA state. Unfortuneately I will not make it back there this summer. I have done about a dozen different hikes there on the same order of difficulty as climbing something like Mt. Washington or Mt. Algonquin in the East. Here are some of my favorite day hikes(non technical walk up summit opportunites):
Hidden lakes peak and lookout (do the "trailless" peak if you do this hike), just follow the ridge up to the left from the col where the trail bears right to the lookout, alot of boulder scrambling with A+ views of Eldorado peak, Sahale, Mt Baker and Glacier peak and Mt Rainier in the distance. About 6-8 hours and 3500 ft elevation gain.
Sourdough Mt, gorgeous north cascade views, about 5200ft elevation gain but the trail is easy on the feet, alot of switchbacks and not rocky like eastern trails.
Hannegan peak (~12 miles round trip and 3000 ft vertical gain), great glacier views of the north face of the ridge north of Shuksan, great Shuksan & Baker views and other of the north cascades on into Canada.
Mt Aix east of Mt Rainier NP, again about 8 hours and 4000ft elevation gain with great vies of Rainier, goat rocks and Mt Adams and the peaks around bumping lake. More arid environment as it sits east of the cascade crest, in the "Rain shadow" of Rainier. Alternatively Shriner Peak within the NP. Another much easier hike in the NP are Sunrise to Burroughs Mountain and Plumber peak near Paradise(Reflection lake parking lot).
Mt Townsend: about 3000ft elevation gain, about 4-5 miles one way. Olympic pennisula near Port townsend, great Olympic Mt and Puget sound views.
If you have 2 days there is Mt St Helens, an extra day is required to get your permit (unless you do so in advance) from a country store and then get to the "climber's bivuac" (really a parking lot with tent sites next to it).
I recommend getting one or more of the Mountaineer's Hiking Guides (x Hikes in North/South Cascades/Washington State).
Have a great time and I wish you clear skies to enjoy the scenery.
John
Caveate: I have done all of the above in Late summer, so snow conditions of June could change the difficulty of some of the hikes. Check conditions at local ranger stations.