You can always go shopping at the malls, but the best place to do your barganing is at the local bazars or souks. I have been enjoying all the different ones in Kuwait. Here you can find anything from gold, watches, material, to electronics. Bargan your heart out.
Quick tip: as someone who may like a tourist, the price will almost always be doubled, so don't buy on the first price. Try to get down to about half that, more than likely, it's what it truly does cost a local. Address: Fahaheel (best), downtown Kuwait, and in SamilyaDirections: Off the main Gulf road, right along the ocean