I lived in Korea for 15 months back in 2010-2011, not in Seoul but spent plenty of time around there.
What area are you staying in? Seoul metropolis takes up a ton of land so it can take 2 hours to get from one side to the other.
Take the subway, it's great and cheap. T-money card is how you pay.
Myeongdong Kyoja (Main Restaurant)(address: 29 Myeongdong 10-gil, Myeongdong 2(i)-ga, Jung-gu) is my absolute favorite place. It's a garlic-based restaurant on the 2nd floor in Myeongdong district (which has lots of shopping and restaurants, the most "Manhattan" part of Seoul. They only serve 4 things, 3 types of soup and garlic dumplings, the kimchi is also garlic-infused.
Another cool district is Insadong, which is somewhat historic area has lots of great little shops and restaurants. You should also get some street food, it's all pretty great (except I didn't really like the dried squid). Itawewon is also a district right outside of the Yongsan American base. It has many non-Korean bars and restaurants as well as late-night street food).
If you love hiking, October is the perfect time to visit. A hike up to Namsam tower (the big tower in the center of Seoul) is great. A little further out (but accessible by subway) is Dobongsan, a great hike with amazing views. Lots of restaurants on the way up and down.
Doing a tour of any of the palaces or old city gates is also a good idea. If you like museums, the Korean War Museum (not just the 1950s conflict but all wars Korea has participated in) is extensive.
Happy to give any other advice, just let me know what you like or don't like to do.