Well... my favourite places to shop in South Korea (Seoul) are Myeong Dong and Dong Dae Mun (after midnight... less people, and it's open till 5am). If you are wanting souvenirs they are best bought at Nam Dae Mun, and you can bargain your prices a bit there, and you can get anything under the sun!
For more arty, and better made gifts head to Insadong, which is an arty traditional street in Seoul, magical place with some great little tea shops hiding off the main strip. However the gifts here come with a price, possibly some pieces also hold guarantees or authenticity certificates.
Souvenirs in Korea can vary, depending on how much mney you want to spend. Some nice gifts are hand made lamps and lanterns made out of paper, yet they look wooden. Or some handmade pottery tea sets... otherwise for cheaper options you can get a million gazillion mobile phone accessories, or key rings for a great price.
My friends always ask for noodles as souvenirs haha and kimchi.
Hand crafted masks are great too, they come in various sizes, and are fairly easy to cart around, more often than not they are wooden and carved.
For Taiwan, the best places in Taipei to shop are XiMen (West gate??) and XiLen (im not sure if thats the correct spelling, but it's a night market...)
XiMen is the 'young' peoples area, really cool, and some unique little stores, you can get pretty cheap tattoos and peircings there too... as for souviners, I guess you could find stuff there... my friends just wanted socks and stuff, so im not very clued up on the 'traditional' souvenirs. Shopping in Taipei is very fun!!!!
Taiwan is handy too, because if you have a cellphone with a SIM card, you can use it in Taiwan, just go into one of their phone stores, and its about $13 but includes that much in call time too.
Korea you would need to rent a phone, if you felt you needed one, or if you have friends, im sure they will have one lying around that they could give you for your stay. We have many in our house, not sure how many actually work.
Hope this was of some help to you!
I love both places, so great! The people, the food and the shopping! Enjoy your trip!!!