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Bear canisters are required in some areas, but the ranger stations rent them out for free. We've had a historically low year for snowpack in the mountains, so my guess is by mid June, most high trails will be snow-free. Water is abundant in June. Washington Trails Association trail reports are your friend.
-Paradise at Mt Rainier, lots of moderate trails in the area. Go up to Camp Muir for better views and a bigger challenge.
-Sahale Arm in the North Cascades. Need a permit to camp, bear canisters provided at the station.
-Check out a lake hike. My favorites are Lake 22, Blanca Lake, and Lake Serene.
-Grand Park in ONP is pretty cool.
You can't go wrong with any spots, but Mt. Rainier continues to blow my mind.