I agree. Cabs in Seoul are very cheap (as opposed to say, Tokyo or the U.S.) but your best way to see the city is through the subway. I think the area by City Hall is very beautiful. There are various palaces in this area situated right next to modern buildings.

Personally, my favorite part of living in Seoul is just the different culture here. Things like eating and drinking soju at a pojangmacha (tent restaurant). Going to a jjim jil bang (kind of a spa/sauna type of thing).