Here were my favorites along those lines:
Seoul: worth a few days at least, with lots to see and do in great neighborhoods. I found Itaewon too full of foreigners for my tastes, but the food options are indeed good. There are tombs and temples scattered throughout the city, but with a map they are really easy to get to. If you want ancient culture, you can take a day or overnight trip to Ganghwa, an island not too far from Seoul, known for it's concentration of dolmens.
For the DMZ Tour, book a few weeks in advance, preferably through Koridoor, the official USO tour company. It's cheaper and better. I was able to slide in last minute, but better do it in advance if possible. The JSA is incredible.
Andong: Hahoe Cultural Village is a really nice spot, and less touristy than you might imagine. It's a bus ride away from Andong city and is at least a half day trip, but it's completely worth it. Really scenic area, too.
Gyeongju: this town is a gem, chock full of ruins and culture, specifically tombs and temples (Bulguksa and Seokram Grotto are a beautiful half-day from town). You need at least two full days here, maybe three. Of all the places in Korea, I think this is the capital of ruins and culture.
Busan: cool city, for sure. Beomeosa is a nice temple in the middle of Busan, but pretty reconstructed. Still liked it better than Yonggungsa, a very touristy temple next to the ocean also in Busan. In the Dongnae area there's a fortress and hiking trail that goes from Beomeosa Temple, so if you like hiking you should definitely look into that.
Specifically for Buddhist temples, I found this site really useful. The problem with many Korean temples is that they're reconstructed from being burned down by invading forces, so most aren't very old. Of all the temples I visited (and I'm obsessed with temples), these were some highlights:
Buseoksa: This is my favorite temple. Big, beautiful, scenic views, and far less touristy than most. It's also home to two of the oldest wooden buildings in Korea, so has a very old, authentic vibe to it. Amazing. The downside is that it's a long bus ride outside of Yeongju, which is a city close to Andong. It was tricky to get to, but completely worth it.
Haeinsa: Home of the Tripitaka Koreana, used to make woodblock prints of Buddhist scriptures. Beautiful temple, too, though it's still a long bus ride from the city of Daegu. Another temple is Donghwasa, closer to the city - lovely temple, though not as impressive as Haeinsa. Daegu city was pretty unimpressive.
Bongeunsa: Great temple with a nice vibe even in the middle of Seoul, worth an hour or so. Very easy to get to, and not too far from some of the royal tombs. Gift shop is great for souvenir shopping.
Enjoy Korea, it's awesome!
