Kuching is probably the best place in Sarawak to shop for souvenirs. The main concentration of shops is along the Main Bazaar Road. Unfortunately, a lot of the souvenirs appear to be mass manufactured items that you can just as easily buy in Thailand or Indonesia, but there are some interesting things to be found. In addition to the bazaar, you'll also find a good selection of real local handicrafts in the Sarawak Museum shop housed in the Tun Adbul Razak Hall.
Some of the 'real' local products include pepper, woven baskets and wood carving. They not only sell whole peppercorns and ground pepper, but also make a lot of pepper items, including candy, coffee and chocolates. Another local food specialty is kek lapis or layer cake. They come in a rainbow of colors and patterns. You'll even see cakes made to look like watermelon! However, since they're full of artificial colors and flavorings, you'll likely find they look better than they taste.
There are a few shopping malls around Kuching, although most of the centers near the main tourist area are a bit old and run-down. The newest mall in the area is the Green Hills Mall in the base of the Pullman Hotel. There are several casual dining outlets here, which may make it a good place to grab a quick meal.
If you're in Kuching over a weekend, there's a weekly market off Satok Road west of the old part of town. Although called the "Sunday Market" it gets started on Saturday afternoon and runs until Sunday afternoon. The open air market sells just about everything from food to clothing and everything in between. Although there are shade-providing tents, it can still be very hot, so be prepared with comfortable shoes and lots of water. You can reach the market on foot, but this is one place it's usually easy to find a taxi.