Korean BBQ is excellent and probably the easiest way to be introduced to Korean cuisine.
I just wanted to highlight some of the foods that I love that you might not get to try if you only stick to the basic Korean BBQ fare.
Soon dooboo (aka hot tofu bowl, aka soon tofu) is basically a spicy stew consisting of lots of tofu, your choice of meats, and some random vegetables. It comes served boiling hot in an equally frakkin' hot bowl. At this point, you can opt to crack a raw egg into the stew (or not). You can choose to get it mild or really spicy -- mild should be safe for most people. Spoon out parts of the stew into your rice bowl, eat together, enjoy. This is close to the best food to get at 2 AM in LA on a weekend (eff pho). If you're in LA, the standard place is
BCD Tofu. My family knows the owners, so buy lots of tofu.
If you like the soon tofu, you may want to try some of the jjigaes (stews, pronounced chi-geh) mentioned by other posters, like doenjang jjigae (soybean paste stew, relatively mild, mmmmmmm) and kimchi jjigae (kimchi stew, spicier).
Finally, my favorite Korean dish:
naengmyun (nang-myun). There are two basic dishes, mul naengmyun (cold buckwheat noodles in a cold, tasty broth) and bibim naengmyun (cold buckwheat noodles mixed with some spicy pepper paste and vegetables). I'm a huge, huge fan of the mul naengmyun -- it comes with some brilliantly fresh-tasting pickled vegetables, and half a boiled egg, all in a refreshing broth. Lots of places will put ice cubes in it to keep it cold. I eat it with some vinegar and hot mustard paste. It is [censored] excellent.
TIP: there's a reason why a lot of Korean joints will have a galbi + mul naengmyun combo -- it just goes hand-in-hand. Something about the contrast of hot BBQ beef and the cold noodles just works. Also, the buckwheat noodles aren't soft so they can be a bit tough to chew. The trick is to chew just enough to allow the noodles to be swallowable, then just go for it.