Cali's Natural Foods - 389 West and 1700 South
Brought to you by the good folks of Sage's Cafe and Vertical Diner, it's the down-to-earth shopping version of the restaurants: high quality, organic, and often local. Unfortunately, also like the restaurants, much of the product is on the pricey side. It is a convenient stop if you are headed to Costco, which is nice, and there are certain "bargains" here too, especially when they have local organic potatoes in stock - they have carried what are undoubtedly the BEST Yukon Golds I have ever tasted!
What I always buy here: bulk staple items such as red lentils and black beans, and one of my favorite treats is the un-sulfured dried baby pineapple!
Shop N' Go - 600 East and 300 South
(Also known as the "Quik E Mart") Indian and Pakistani ingredients and prepared meals, at very good prices. They also have a small section of produce, such as eggplant, peppers, okra, and cilantro...
What I always buy here: frozen samosas and naan, "instant" shelf-stable curries, and a hot samosa to go!
SouthEast Supermarket - 422 East 900 South
(Formerly known as "Tay Do Market") Probably the largest Asian market in SLC. They have everything you need for Japanese, Chinese, Vietnamese, and Thai cooking, and a fair amount from other locales such as India, both food ingredients and equipment (I bought a Laotian basket steamer here to make sticky rice!) Their prices are always less than the chains on the items you can actually get at the chains (i.e. Sriracha sauce) and there's just so much more to choose from. They also have a fairly large produce section with some things I've not quite identified yet, and a frozen section of mock meats that is at the very least entertaining...
What I always buy here: nori for sushi, noodles and rice paper for spring rolls, fresh mint and bean sprouts, and staples such as rice vinegar, soy sauce, and rice.
Super Mercado De Las Americas - 1179 South Navajo Street (~1350 West)
Off the beaten path (for me anyway), this place is a bit tricky to find. When I want to make tamales or a giant batch of Chile Verde, this is where I go. The produce section usually has the best deals (and freshest product) in town for "Latino" items such as jalepenos, limes, onions, cilantro, etc. They have fresh tortillas and baked goods; I'm pretty sure they are made with lard so I haven't eaten them, but they make the place smell heavenly!
What I always buy here: masa harina and dried corn husks for tamales, giant cans of hominy for Chile Verde, and dried chile peppers of all varieties.