When I was living in Vancouver and didn't have a car before car2go launched, I used to stay at the Mediterranean Inn when I visited Seattle. It's conveniently located in the Lower Queen Anne neighborhood, where Seattle Center is located.
There are Festals at Seattle Center to celebrate different cultures. There's food and usually some type of singing and dance performances throughout the Center the weekends that they're on:
http://www.seattlecenter.com/festal/
The Woodland Park Zoo is well worth a visit, and they have concerts during the summer.
Although you can get close by transit, almost no one ever does, so if you have a car, I would suggest the Kuboda Garden:
http://www.kubotagarden.org/
Food wise, you can drive or take light rail and then walk 10 minutes to the small downtown in Columbia City, where one of the best bakeries in town, Columbia City Bakery, is, and across the street is wonderful thin crust Neapolitan pizza restaurant, Tutta Bella. (They have at least two other branches in Seattle, too.) Also there is a Green Eileen shop, where you can get second-hand Eileen Fisher clothing at 20-25% of the original cost.
My favorite gelato places, at least when I could eat it, are The Fainting Goat on 45th Street in Wallingford, not that far from the Stone Way Tutta Bella, and D'Ambrosio on Ballard Avenue, convenient if you head to the Ballard Locks. Most people go to the Locks and turn around without continuing up the road, where you'll find Ray's Boathouse -- Ray's Cafe upstairs is cheaper, and if you're lucky, you can get deck seating -- and even farther, a lovely walk along the docks, where people moor their boats. Even farther up is Carkeep Park, which is on the water and has a beach.
Again, easy with a car, is Alki Beach in West Seattle, and in the town center of West Seattle, bakeries and restaurants. That's generally from where they get the shots of downtown Seattle.
One of my favorite things is taking ferries. From downtown, you can ferry to Bainbridge (35 min each way) or Bremerton (60 mins each way) -- you pay in Seattle. You can just turn around and come back. With a car, we've driven to West Seattle, had lunch, driven to the Fauntleroy Ferry Terminal at one end of West Seattle, take the car ferry to Vashon Island, gone to Farmer's Market there, driven through Vashon to Tahlequah, taken another car ferry to Point Defiance, where there's a park, and go to Tacoma, which has a great museum district: Museum of Glass, if you can't get enough Chihuly, Washington State History Museum, America's Car Museum, and the Tacoma Art Museum.
http://www.wsdot.wa.gov/ferries/
http://www.traveltacoma.com/things-to-do/museum-district/
One Seattle Museum that not enough people go to is the Museum of History and Industry (MOHAI), which is now more conveniently located in South Lake Union:
http://mohai.org/
The Olympic Sculpture Park is at the border of Belltown and Lower Queen Anne on Elliot Bay. It's free, and there's a lovely waterfront walk there as well:
http://seattleartmuseum.org/visit/olympic-sculpture-park