Mumbai is a shopper's town: in the same day, you can sift through alleys full of antiques in Chor Bazaar, haggle for trinkets on the Colaba Causeway, and stop in at the Brioni showroom at the Taj Mahal Palace Hotel for marked-up luxury goods (though we'd recommend you get your Chanel and Armani back home to avoid the huge import taxes).
The Causeway, Kemps Corner, and Breach Candy are all trendy shopping areas in South Mumbai; the latter two are chic and pricey. A walk down Colaba Causeway will probably take you past most of the things you want to buy in India—shoes, clothes, cheap knickknacks, cheap cotton clothing, jewelry, and wraps—displayed at stalls lining the road; more expensive items are found in the air-conditioned shops and boutiques behind the stalls on this same road.
The arcades in top hotels offer a little bit of everything for a lot more money than anywhere else, but the merchandise is beautiful and the pace unhurried (and it's climate-controlled). If you're looking for the kind of stuff you can't get anywhere else in the world, and a more vibrant experience, throw yourself into the middle of one of Mumbai's famous bazaars. After all, odds are you didn't come to India to visit the Louis Vuitton boutique.
The city's department stores are good for one-stop shopping, and Fabindia and the Bombay Store both have a large number of branches in the city.
Throughout Mumbai many smaller shops are closed on Sunday (some of the suburbs are closed a different day: in Worli, up to Bandra, they're closed Monday; and in Bandra, up to the suburbs, they're closed Thursday, although many areas are also in the process of switching to Sunday). Malls, however, are open every day. They are especially crowded on the weekend (mall-gazing—that is, large-scale window shopping—has become a new Mumbai leisure activity).
Once you've exhausted Mumbai proper, you can venture out to the suburbs, where prices tend to be lower and the malls more numerous. Linking Road in Bandra is a trendy place to shop, and Juhu's main strip, Juhu Tara Road, is lined with cutting-edge new boutiques, shops, art galleries, and restaurants.
Some good and cheap Mumbai buys: silver jewelry, handicrafts, handloom cotton and silk clothing and household items, eyeglasses, DVDs, CDs, and books.