You can get information about traveling to and from the airport along with some travel basics here
I spent 5 days in Seoul it’s a huge city of 10 million people with many things to see and do. This is just a brief overview of what I did while I was there.
Getting around
There is a very good subway system in Seoul. The station names are listed in English and Korean, and at the stations they post the name of the last and the next station the train is going to. You can get a map of the whole subway system. You pay using a refillable card which you can buy, and refill at machines in each station. When you pay you need to select your destination. I had some difficulty figuring out where I wanted to go before the machine timed out, but I eventually got the hang of it and every time I looked lost by a machine a helpful local stopped to try and help.
Google maps does not work well in South Korea. Apple maps is not great, but it is better.
Where I stayed
I stayed in Seoul at the beginning and the end of my trip. I stayed at the Kensington Hotel for two nights on arrival and at the Nine Tree Premier Hotel Myeongdong 2 and the 24 Guesthouse Myeongdong Avenue at the end of the trip.
The Kensington Hotel and the Nine Tree Premier Hotel are both very nice hotels. The 24 Guesthouse was relatively cheap and basic, but it was clean and I needed a place to stay close to the start of the Seoul 100. I would recommend the area around Myeong dong as a good central location with lots to see and do, but the whole city is well connected by public transit.
Things to do
My list is a more a list of what I did on my short visit you can find a more complete list here https://english.visitseoul.net/index
The largest of the royal palaces built during the Joseon Dynasty. As with several of the other main historic sites Korean tourists rent traditional clothing to visit the palace (you can too). There are more places and temples to see in Seoul.
Bukchon Hanok Village
A mixture of small cafes, store and traditional Hanok homes located just around the corner from the Gyeongbokgung place. Definitely worth a stroll.
A large traditional market. With food and household items.
This massive market has outdoor stalls and indoor vendors. It is the largest traditional market in Korea.
Located on the fancy shopping street Myeong Dong. The place fills with carts selling food of every type at 5:00pm. A street food lovers dream.
Held Banpo Hangang Park along the Ham river. which is worth a visit. It’s only open on Saturday and Sunday nights so we missed it.
There are some beautiful mountain trails and hikes right in the city. Including the Baegundae-Ui Gugok trail to the top of Baegundae Peak. I ran around several of the trails as part of the 50K at the Seoul 100. Great views but you have to work for them!
Seoul has a great, long cycling route running through the heart of the city along Hangang River. You can also ride back out to the airport! The route is traffic free and well sign posted.
In front of city hall it’s a local gathering place. When I was there, there was a free outdoor public library where you could hang out in chairs and read books. You could also rent yoga mats.
Overall Seoul is a big city. It’s easy to walk around and the subways are great. It’s very safe and clean. I enjoyed poking around and you will too. There is lots to see!
Racing and Traveling over 50. 
That looks very interesting. We do not see much about visiting Korea. Thanks.