Shiney, there are so many nice hikes all over that I would say the best thing is to figure how far you want to drive. For example, if you live on the eastside, the trails that Whalcamole mentioned are good and close.
One of my favorites is a pretty long hike and a long drive N of Seattle. It crosses the Cascade Crest and drops you down into Lake Chelan. Not a day hike. LIkewise, for overnighters lots of great trials in the North Cascades Wilderness. This has been a dry year, but still for some of these you might want to check snow conditions. Off HiWay 20 to the north are lots of triails mostly made by stream fishermen on the south side of Mt. Baker.
About an hour north of Seattle in decent traffic is the drainage of the Stilliguamish and Sauk rivers. Many good trails here, and some get pretty high pretty fast.
People that live in Tacoma like the area around Mt. Rainier.
I used to live on the Peninsula. Really nice trails on the south side of Crescent lake, as well as Hurricane Ridge. Also out there are the great lowland trails aong the Hoh and Bogachiel Rivers, as well as lowland hikes to Lake Ozette and on to the Beach.
I would advise picking an area, and then stopping by the library to borrow any one of many very nice guides that are specific to small areas. Most of these books will also mention if there is exposure on a trail, as not everyone is comfortable with that. Also if it is something where you should have an ice axe.
Ha