I was at Camp Hovey in 1989-1990, so my info is dated and I'll let the guys with more recent experience answer those questions.
Camp Casey backs up to Soyu-san, or Mount Soyu. There is a 17th century Buddhist temple on the other side, I highly recomend visiting it, it's beautiful. I have about a hundred pics of the place. If you like hiking and mountains you'll have fun, I've hiked all over near Casey 39'er outside of Hovey.
If you can take a Korean language class, they used to have them on post. Even a few words of Hangul (Korean) spoken with a smile will get you a lot of mileage with the locals. Be repectful of them and they will treat you well, in my experience. Korea was one of the best experiences of my life. Of course, I did a DMZ tour as a grunt, and was lucky to have excellent officers and senoir NCO's in my company and Battalion.
But take advantage of any chances you get to see the country-there used to be tours you could take that left from Camp. The folk village, ski trips, etc. I wish I had done more of that and less drinking and whoring (although I have no regrets in that area). And groove on the local food, it's quite interesting fare-there are more kimchis than just cabbage.
Have fun, with the right attitude it's a good tour.