African Restaurant & Market - 1878 South Redwood Road
Surprisingly, this is *not* the only place serving East African food in SLC, but it is certainly the most vegetarian-friendly and what I call "gringo-friendly" since the other two places had trouble answering my questions about whether the food was vegetarian. I've been going there for years, in fact since before it transformed into mostly-restaurant (instead of mostly-market). Reliably tasty, and they supply you with plenty of budeena/injeera to savor your Ethiopian stews with. I recommend the veggie combo, and when I've gone with meat-eaters they tend to like the wat/combo plates as opposed to the rice-based dishes. You can read more reviews here on Yelp: http://www.yelp.com/biz/african-market-and-restaurant-salt-lake-city
Cafe Solstice - inside Dancing Cranes Imports at 673 East Simpson Ave (2240 South) http://cafesolsticeslc.com/
Full disclosure: this is where I work part-time! But even if I didn't, I would recommend it...it has everything I love: coffee, tea, and other drinks alongside a mostly-vegan/veg light lunch fare menu and yummy treats. My personal faves are:
Drink: Soy "Dirty Devi" - A Devi Chai with two shots of espresso
Sandwich: Hummus and Olive on multi-grain. I've never been an olive fan, but Erin's tapenade has changed my mind about olives!
Meal: Solstice Tacos and Side Salad with Creamy Cucumber Dressing. When I read the description, I wasn't impressed - but HOLY WOW. These nontraditional tacos are addicting!
Muffin: Cherry White Chocolate. I don't seem to work on the day Erin bakes these - I always get them as day-olds and they are the bomb!
The Chaat House - inside Qaderi Sweetz and Spices at 3546 S. Redwood Rd
This place is a bit of a mystery - I haven't quite figured out what the hours are for the restaurant portion of the store. Weekends are a good bet, as long as it is before 7:00 p.m. They feature fast Indian-Pakastani food and treats at great prices. The reviewers on Yelp seem to focus on the Chaat, which I haven't actually tried, but the combo dishes are quite delicious!