Parco (パルコ)
Not too overpriced. Two floors for guys, and the rest is primarily for ladies. The top floor has a cool knickknack store called Vagabond.

Tsuruya (鶴屋)
Get ready to break the bank. Name brand, name brand, and a little more name brand. They have clothes, cosmetics, jewelry, cookware, and much more. In the basement they have food shops where you can get hard-to-find foods for a premium.

Don Quijote!!!! ドンキホーテ
Anything and everything you could ever want, from food to electronics to S&M costumes to Heinz Ketchup to car wax to liquor to the Shake Weight.

Hamasen Youme Town Mall (はませんゆめタン)
This is a decent sized mall. There is an international food store where you can buy ALL kinds of food, including dill and Thai curry. There is also a Subway in the mall.
Google coordinates: 32.771308,130.72664

Eon Mall Kumamoto Clair (イオンモール熊本クレア)  
Big-ass kick-ass mall. Do it!
Google coordinates: 32.739566,130.74357

Canal City (キャナルシティ)  
A big big mall in Fukuoka. Worth a visit if you like malls and like to shop.
Google coordinates: 33.589688,130.41074

Tenjin (天神)
The downtown shopping area in Fukuoka. This also has night life.
Google coordinates: 33.591082,130.399153

Costco (コストコ)
Big sized food! It’s in Fukuoka.
Google coordinates: 33.652244,130.486979