1. Tours of the Georgetown Steamplant are both free and amazing. Link -->
http://www.georgetownsteamplant.org/. The openings are limited, but might line up with your trip.
2. The top of View Ridge (around NE 65th Street @ 50th Ave NE) has incredible views of Lake Washington, Mt Rainier, and both the Cascade and Olympic ranges. Start at the bottom of the hill, and feel like the king of the world when you reach the top.
3. University Village shopping center is fun for people watching, if you're into that sort of thing. Tonnes of shops to pick up a cheap coffee and snack, then watch people brows the high end shops.
4. DON'T RIDE THE DUCK BOATS. Them things ain't safe. I work in the maritime industry, and those amphibs do not have a good safety record.
Cheap eats that are unique to Seattle:Beth's Diner
The 5 spot (also a diner)
Blue C Sushi (conveyer belt, and lots of fun. Cheap as long as you choose dishes correctly)
Top Pot (already mentioned, but I wanted to reinforce!)
Din Tai Fung (dumplings. Not overly cheap, but amazingly good)
Hales Ales (one of the first micro-brewerys in Seattle. Economical during happy hour)