As with many of its geographical neighbors, Korea is not a good place to order dinner by closing your eyes and pointing to something on the menu. What's considered edible here will often churn the stomachs of all but the most hearty foreigners. The locals gobble down bumpy sea urchins, insect larvae, and unidentifiable roots and fungi without a flinch, while smelly dried squid and rubbery octopus are considered drinking snacks here--the Korean equivalent of peanuts or nachos at the bar. One fear that is unfounded, however, is that Rover will end up on your plate. Yes, some people do eat dog here (as they do in China, Vietnam, and parts of other Asian countries), but only at specialized restaurants--at these places, that's all they serve.
In fact most restaurants in Korea are narrowly specialized places. One restaurant will serve only kal guk su (fresh noodle soup), while one next door may only serve fish. For obvious reasons, locals trying to host you in Seoul will want quite specific answers when they ask you what kind of food you'd like to sample. Restaurants that cater to lunch crowds serve a slighter greater variety, largely because there are a dozen or so set dishes that nearly every place serves during the day. The most common are probably pibimbap (rice and vegetables with a spicy sauce and a fried egg), kimchi chigae (a spicy stew with kimchi and pork), and soondubu (a spicy tofu and clam stew). The Koreanized Chinese restaurants feature fried rice, chopchae (stir-fried clear noodles, vegetables, and pork), and jajong myon (noodles, vegetables, and meat in a dark brown soy gravy). If you find a few standards that you enjoy, you'll be able to find a place to eat nearly anywhere.
There is a way, however, to try a wide variety of Korean food at one sitting: by going to a restaurant that serves han jong sheik, a Korean banquet meal. The simpler versions feature one or two main dishes and wide variety of side dishes. The most elaborate try to simulate a traditional royal meal, with thirty or more small bowls of different meats, seafood, soups, and vegetables. Korean restaurant menus in the US, of course, will generally offer more of an overview than you would normally find in a single spot in Seoul.