If you go to the University of Washington, up and down and right around the Ave is full-to-pretty-much-bursting with all kinds of cheap, good food, as well as some cheap, bad food.
For the good stuff, try Aladdin Gyrocery, Araya's Place, Chili's, Guanaco's, Jewel of India, Korean Tofu House, Nook, Pam's Kitchen, Samurai Noodle, Taste of Chicago, and Thai Tom. If you don't go to the University of Washington, you can still eat at all these places—those Huskies don't own them. But there's also a whole city of strange, delicious, inexpensive food out there—go get some!
If you've never had dim sum (or even if you have), go to weekend brunch at Jade Garden in the International District with some friends. It's a crazy mob scene in the entryway; get your name on the list, and be prepared for a worthwhile wait. Once you get a table, you choose all kinds of dumplings and noodley things and you-know-not-what from carts-on-wheels that wheel by. The cart-wheeling ladies are brusque, but don't be afraid to say yes to lots of stuff—it's all remarkably cheap, and the more you test out, the more you'll like.
For breakfast, lunch, or dinner, you must try Cafe Presse. It is French, but crowded-and-fun-French instead of snobby-fancy-French. The menu is fairly explanatory, they'll tell you more about anything that sounds odd-but-possibly-good, and they really don't care if you can't pronounce things right. Lots of stuff is under $10—try a soup, and/or a croque monsieur (that's French for "extra delicious ham and cheese sandwich"), and/or a salad—it's all so, so good. (Note that "steak tartare" is French for raw meat. Try it! But with a friend—they give you a ton.)
Salumi in Pioneer Square has sandwiches that are literally world-famous. They cure their own meats, and they have weird- sounding choices—lamb prosciutto, tongue (it's beef, not people). Go stand in line at lunchtime and get one. No, don't, actually: Take a friend and get two, then go halvsies.
For inexpensive sushi and yummy Japanese hot snacks, hit the happy hour at Japonessa downtown—currently, the all-ages portion goes from 11:30 a.m. to 6:00 p.m. and 10:00 p.m. to close, with additional 21-plus happy hour in the bar from 6:30 to 8:00 p.m.—SO MUCH HAPPINESS! It's kind of swanky-contemporary here, and the raw fish is fresh because they sell a hell of a lot of it. This would be a good date spot; taking someone to a good-looking place with interesting food makes you seem even greater than you already are.
Also a good date, same reasons: oysters on the half shell at the happy hour at Elliott's on the waterfront. Elliott's is fancy but not too fancy, with a sparkly-water Elliott Bay view; they have good chowder and other snacks for cheap at happy hour, too, for the timid. However, if your date doesn't want to try raw oysters, well, what else are they not going to want to try?