Check out Dongdaemun and Myeongdong for shopping. Myeondong will be flooded with people if you go during the day. Gangnam and Apgujeong are more upscale areas with bigger and more expensive clubs.
Party in Hongdae(Mansion, M2, Noise basement), Itaewon(Rococo, Luv, Volume), or Gangnam(Answer, Eden, Heaven). Check out Insadong and Samcheon Dong for art shops, boutiques, cafes and scenic walking. Visit hyehwa if you want to see a theater show and Namdaemun market to pick up all kinds of random souvenirs.
I also suggest that you eat a ton of korean barbecue. Galbi is my favorite. Many of the soups are also delicious. Kamgyetang, Samgyetang and Kimchi jigae are all good. And make sure to drink plenty of soju(mixed with Aloe is great if you hate the taste of it straight).