Try getting up to Mount Rainier. It is about 2 hours to the Park boundaries, maybe a little further. Once you get inside the Park, you'll have to drive a bit further, but the hiking and scenery is next to none.
Check out hikes out of Sunrise specifically. Pure awesomeness.
Take I90 to Exit 20 high point and go up tiger mountain 3. If you want solitude find the Nook Trail. Caution very steep but worth the view. You get the same view as Si with a shorter drive and hike. Try mount Dickerman on the Mountain loop highway or Granite Mountain. These last two are summer hikes unless you wish to travel on snow.
There are enough trails to spend a whole day right over in Redmond. We've spent many hours there, with several different trails over a large area (800 acres). Redmond Watershed Preserve is about 7-10 miles if you hit all of them but longer getting from one to another. Pretty dense forest with some creeks and ponds, nurse logs, trilliums in spring. It's pretty easy with just slight hills.
Directions: Follow east-bound SR 520 to the end, bear left and you?ll end up heading north on Avondale Road. After 1.25 miles turn right on Novelty Hill Road. Drive 2.4 miles to 218th Ave. NE. Look for and follow the Watershed Preserve sign on the left. Check a map and you will see that there are a couple of other access points but this is the easiest to find.
Check out Cougar Mountain in Issaquah. However if you want to go for a real hike. Google Mount Si. near North Bend. Really if you want to get out look into aquiring a bike somehow and bike the Burke-Gilman Trail. It's the best.
Right in Seattle, Discovery Park is pretty good. There's nothing too difficult or long, but there are trails everywhere and a couple beaches. It's a nice local spot that not too many people think about.
As a kid my dad alwasy took me hiking up tiger mountain. It is a moderate hike, better part of the day for a family. Not sure if this is what you are looking for or not. Been a while since I was last in the area but really do miss it.
I don't do much mountain-type hiking in the winter here (muddy, muddy), but I love the Centennial Trail. While there's not a lot of terrain variety, you can bike or walk for miles and miles. Looks like we're in for nice weather for the next few days (and I'm on vacation), so I'm planning to take my dog on 5-to-8 milers every day.