I lived in Columbia for a number of years and still have family there, and there are some very good options to try out, though I would lower your expectations for quality ethnic restaurants. Shakespeare's, which has already been mentioned before is a local institution. The pizza is pretty tasty, the beer is cheap, and the atmosphere is always good. It's also open pretty late, which it sounds like you're interested in. Another local institution, which I would put at the top of my list of places to go is Booches, which is right off the main drag on 9th street. Booches is an old pool hall (well maintained tables) with a small little flat-top grill that turns out some of the tastiest hamburgers I've ever had the pleasure of eating. The burgers are small (though not thin), greasy, and served on a square of wax paper. I could eat them all day long. The beers are cheap cold, and the atmosphere is simply wonderful. Don't ask for mixed drinks beyond a whiskey and soda, the bartenders don't much like foolin' around.
Also downtown is Uprise Bakery, which is a newish addition to the scene. Very good breads, and simple sandwiches for lunch. Tellers is right on the Corner of Broadway and 9th and has simple, fresh, and sometimes creative food and is a very nice place to spend a meal. It's also open late, and is a great place for a drink and snack. Cherry Street Wine Cellar turns out some of the more inspired food in town, and has a very good, reasonably priced wine list to match. India Rasoi is right downtown, and is quite good if you have the jones. Also, Ragtag Cinema is a very cool little spot downtown that has a great calendar of independent movies. They also have a little bar and cafe at the front of the house. For breakfast you can't go wrong with Glenn's.
Away from downtown you can hit the Cherry Street Farmers Market, which is probably going full steam about now. The market has grown substantially over the years and is pretty impressive now. Murray's, which is south of downtown, is a wonderful restaurant/jazz club that serves consistently good, simple food. Also out that way is Shakey's frozen custard, which is as good as any I've had elsewhere.
So that's what I would suggest as some good options for eating out. Of course, there's a lot more there and certainly some good places that I'm forgetting. But I would definitely stand by any of these choices.
If you want to take a trip outside of town, the town of Rocheport, which hugs the Missouri River, is a good destination. There's a little place called Abigail's in the tiny downtown area that has very good food, and you'll pass a winery on your way in that has good food and a great view of the river.
Enjoy