Yu Yuan Mart(豫园商城) is probably the best market for shopping for Chinese characteristic stuffs. It is a huge mart that consists of several department stores and thousands of small shops. Almost anything that is Chinese can be found here: Jade, Antiques, Artworks, Crafts, Souvenirs, Clothes, Jewelry, Tea, Teasets, about everything you can imagine! Must go!
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Other Marts
A.P Plaza (亚太盛汇休闲购物广场)
Qi Pu Road Market (七浦路服饰批发市场), aka The "cheap" market.
Rong Fu Mart (融富百货)

Bargaining is almost essential when you are shopping in Shanghai. Not all, but quite some of the shop owners, especially in tourism places, tend to put a way higher price on tags than the price they can usually sell, in other words, the true value. Local people usually do not even look at the price tags in small shops in a mart like Yu Yuan mart(豫园商城). But foreign tourists would not know this...
A guide to Bargaining in Shanghai

Online shopping is easy and nice. How much feasible is it however to do online shopping in Shanghai?
Read here

Xu Jia Hui(徐家汇) is the best place if you are shopping for computers, electronics, cameras. There are four marts here for computer related stuffs.
Pacific Digital Plaza I (太平洋数码广场一期)
Buynow(百脑汇)
Bestbuy(百思买)
Pacific Digital Plaza II(太平洋数码广场二期)
If Xu Jia Hui(徐家汇) is too far for you, you can still get them in other places.

Middle Huai Hai Road(淮海中路)
CyberMart (赛博数码广场)

Lu Jia Zui(陆家嘴)
Pacific Digital Plaza III(太平洋数码广场三期)
Buynow PuDong(浦东百脑汇)

Wu Jiao Chang(五角场)
CyberMart (五角场赛博)

Ikea(宜家) has a huge mall near Xu Jia Hui(徐家汇) where you can shop for almost everything that you would want to put in your home.
MUJI(无印良品) and TAYOHYA(多样屋) are two smaller chain furnishing stores. They provide expensive and quality furnishing stuffs.
B&Q(百安居) and HomeMart(好美家) are two popular stores for decoration materials and furnishing stuffs.

Shopping in malls and department stores in Shanghai is not much different than in other countries. There are plenty of options in Shanghai.

West Nan Jing Road (南京西路)
During the last 15 years, the area around West Nan Jing Road has been reconstrcted as the new CBD of Shanghai. Now it seems almost like there is nothing but class A office buildings and shopping malls containing only luxirious stuffs. And it is also where beautiful women are most seen in Shanghai.

Plaza 66
恒隆广场
Plaza 66, which has 66 floors, is the tallest building in Pu Xi(浦西) of Shanghai. There is a department store from the first floor to the fifth floor and it is usually regarded as the leader of the fashions in Shanghai

SOGO
久光百货
SOGO department store sits right at the exit 2 of the Jing An Temple Station(静安寺站) of Metro No.2. And it is right next to Jing An Temple. The convenient transportation brings millions of people here every day.

West Gate Mall
梅龙镇伊势丹
While its neighbour - Plaza 66 (恒隆广场) is almost just for wealthy people, West Gate Mall provides a nice varieties of goods for both main stream and the rich.

More on West Nan Jing Road

Middle Huai Hai Road(淮海中路)
For many years, Middle Huai Hai Road has been known as a premium place for shopping and sensing the trends of fashion in Shanghai.

Parkson
百盛
Which is the most popular department store on Middle Huai Hai Road? If you stop some one walking on Middle Huai Hai Road and ask him/her this question, the answer would probably be "Parkson".

Shanghai Times Sqaure
大上海时代广场
There are two Times Squares in Shanghai: this one in Pu Xi and the other in Pu Dong. It is a very spacious shopping mall, which is kind of rare in Shanghai.

More on West Nan Jing Road

Find more malls/department stores.

There are more than 100 HyperMarkets that covered almost every area in Shanghai. And convenience stores can almost be found on every street of Shanghai.

HyperMarkets:
Carrefour(家乐福)
Lotus(卜峰莲花)
Walmart(沃尔玛)
METRO(麦德龙)