Most cities here have at least one large public market. In Kunsan it is next to the train station (Kunsan "yuck" in Korean) and is in a long, covered building.
Here you can find almost anything that is available for food in Korea. From seafood to herbs to dogs and rabbits, it's all here.
Additionally, you can find men's and women's clothing, bedding, cloth, kitchen ware, and hanboks - traditional korean dress.
Even if you don't buy, it's worth a stroll just to get a better idea of the culture of Korea. You'll probably be doing just as much staring as is being done at you. But just relax and enjoy yourself. I guarantee it'll be an interesting experience.
For those who are adventurous, try some of the foods that are cooked or baked in the market. There are several small bakeries as well as a few food stalls here and there throughout the place. If you are really adventurous try "bondaeggae" - boiled silkworm larvae. Personally I can't stand the stuff, but at least I gave it a try.
The main shopping area in Kunsan is in Young-dong. It's pretty easy to find and is across the street from the only KFC in town. Any taxi will take you there and know exactly where you want to go if you say "young-dong."
The street itself had a recent facelift because of the World Cup and is closed to traffic. It is also painted and adorned with the logo of the city, among other things.
Here you can find mostly clothing from baby clothes to high-end business suits. Also, shoes, sporting goods, discount stores, jewelry shops, a couple of small restaurants, and some accessory shops.
The street is about 1/2 a mile long and branches off in a couple places as well. It's very well lit and stays open until about 10 or 11 pm. It's located downtown and is usually packed with people at any given time. Since Kunsan is kind of a small town, it really is the town center and you can find almost anything you need within walking distance - movie theaters, restaurants, bars, etc.
If you go, be sure to stop at one of the outside food vendors for some roasted sweet potatoes or some freshly made snack cakes in the shape of walnuts, bananas, or peanuts. Or you can try the fried squid snacks or the round fish patties. They are all delicious and cost less than a dollar.
Just be careful not to walk too far down or you'll end up in the red light district which can be a strange experience to say the least.
What to buy: In Kunsan it is very difficult to find souvenirs - Very Difficult. So if you go shopping here, it should be for mainly what I've already listed.
Also, if you can't find what you want in or around young-dong, you can always catch a cab to E-Mart - basically the Korean version of WalMart. It's not as much fun as the open shopping area though and, in general, the prices are about the same.