Seattle Center has plenty of things to do to pass some time, but it's all a bit spendy in my mind. A ride up the Space Needle, Pacific Science Center (that one would probably be fun for the kids), Chihuly museum, Museum of Pop Culture, an IMAX Theater, there's a food court there but nothing special.
I should have lots of answers for you but I'm a north ender and mostly avoid downtown Seattle. I'm not much help on the restaurants front.
Strolling around Pike Place Market is probably worthwhile. Not far from there down to the waterfront.
There's the REI flagship store. It's huge. Has an indoor climbing wall, more gear than you can shake a stick at.
Taking a ride on the ferry system is pretty cool. The one from downtown goes to Bremerton or Bainbridge Island. You can drive a car on and then explore around Kitsap and/or Olympic Peninsulas (well, not if you go to Bainbridge, cuz your on an island). Or you can walk on, a lot simpler, and just enjoy the water crossing w/ views of mountains in all directions. There are numerous ferry terminals that depart to different destinations.
Speaking of islands, Puget Sound is full of them. The San Juans are gorgeous but a bit of hassle, it's recommended to get ferry tickets in advance and would be quite a bit for a day trip (not impossible though). There are numerous islands in the chain but San Juan and Orcas are the most developed. Whidbey Island is more doable for a day trip. You can catch a ferry to the south end of the island (Mukilteo to Clinton) or drive onto the north end via Hwy 20. The bridge over Deception Pass is quite spectacular.
I might think of more things later. Like I said I don't go into downtown much. To me the best thing about Seattle is everything that surrounds it. You have the right idea going hiking around a couple of the big peaks. Enjoy!
(Do you have an itinerary set for hiking? I haven't done all the trails around Rainier but I've hit several of them. Be prepared for crowds, Rainier on a nice summer weekend gets crazy.)