I am a Navy wife and we were stationed there from 2000-2002. Assuming it's still there, ride with someone and find out how to drive to the Engineering base. This is the one drive you really need to learn. From there you can safely park and walk to several markets (Tongdaemun) and or use the subway and really have fun. Either an umbrella stroller or a back pack is best for your son. I used both as mine were 6 months, 2.5 and 4.5 when we got to Korea. Eventually, I found a good oshima to help with the house and mostly watch the kids so I could go out. Others like Namdaemun market better for it's variety. This one you don't really drive to a bus or train is best.
If you're not ready to drive then get to the back gate and walk to the subway station. From there you can go anywhere. Start with Keungbokgung. It's a great palace and easy to get to using the subway and a short walk. I don't remember the names of the stops and all, but someone there will help with that. That palace (gung) is at the end of a main street in Seoul. On that same street is a cultural arts center and an opera house. They have different shows all the time. It's free. When I was there they had an extraordinary display of quilts. The Koreans really have taken American style quilting to the next level. Right outside the front gate is the War museum. It's pretty amazing, don't miss it just because it's so close. There was a new national museum being built when we lived there. You could see it from Blackhawk village where we lived. I'm sure it's open now and probably well worth a visit. In the vicinity of that new museum was a sculpture park. It should still be there and would be fun with the kids.
As far as shopping goes it can be very overwhelming. I would learn where the markets are and browse your brains out for a few months. Jot a few notes to yourself where you saw various types of things. If you don't it will all get jumbled up for sure. But if you see just the right thing for someone back home, get it. You have to ship by mid October to get things home by Christmas. A few of the markets I enjoyed, flowers (there's one a short bus ride across the Han River and one a short drive south on the freeway), shoes, toys & stationery, fabric, and of course clothing as far as the eye can see. Shopping for ceramics locally wasn't ever easy for me, but someone will know where to go. I went out of town to some of the ceramic villages. This is a trip you should plan on going on at least twice in your time there.
Iteawaon market is a good market to see the wide variety of what is available and to aquaint yourself with what a high price is. This market is designed for those that don't have the time or inclination to explore beyond what's right outside the gate. You can bargain there as everywhere. Learn the language of numbers after that of manners.
Of all the stuff I bought while living in Korea for 2 years the only purchases I was ever sorry about were products from China. I don't care how good they look and how cheap they are, they fall apart within a year. The only exception were the pearls.
There's a fun place to visit where the old Korean lifestyle is on display. I think it's just called the Korean Village. It's not hard to drive to, but I think there are tour buses that go too. That was very memorable for us. It's a whole day and worth it for the whole family.
I have to say we absolutely loved our time there. We loved the Korean culture. The people are warm and generous. As long as you return the favor you will love your time there. My only regret is that I didn't spend more time learning the language. It's not that hard no matter how foreign it may seem. It's completely phonetic as is it's alphabet. It's worth the effort and most Koreans you meet will be patient with you when you try to speak in their tongue.
Don't be afraid and jump right in. It'll be great!
K. Harris
(Currently starting a new adventure in Naples, Italy)