Lamb's (169 S. Main St., 801/364-7166, www.lambsgrill.com , 7 a.m.-9 p.m. Mon.-Fri., 8 a.m.-9 p.m.Sat., $15-20) claims to be Utah's oldest restaurant; it started in Logan in 1919 and moved to Salt Lake City  in 1939. You can still enjoy the classic 1930s diner atmosphere as well as the tasty food. The menu offers seafood, steak, chops, chicken, and sandwiches; Lamb's is an especially good place for breakfast.
For moderately priced food and history of a different sort, try The Lion House (63 E. South Temple St., 801/539-3257, 11 a.m.-8 p.m. Mon.-Sat., $8-18). Built in 1856, this was one of Brigham Young's homes, where his 27 wives and 56 children spent most of their time. High-quality cafeteria-style meals are available for lunch and dinner in the basement dining room, formerly the household pantry.
If fast food is more your style, then at least try the local purveyor: Crown Burgers (downtown locations at 377 East 200 South and 118 North 300 West, 10 a.m.-10:30 p.m. Mon.-Sat., $4), with many outlets throughout the Salt Lake area, is the local favorite for char-grilled burgers and good fries. For that special Utah touch, ask for "fry sauce" with your French fries—it's a local condiment that's remarkably like Thousand Island dressing without relish.
Two excellent options for barbecue and soul food are Sugarhouse Barbecue Co. (2207 South 700 East, 801/463-4800, www.sugarhousebbq.com , 11:30 a.m.-9 p.m. Mon.-Thurs., 11:30 a.m.-10 p.m. Fri.-Sat., 1-8 p.m. Sun., $9-15), with dry-rubbed ribs and pulled pork, and Rib Alley (533 South 500 West, 801/359-9926, http://riballeyslc.com , 11 a.m.-2 p.m. and 6-10 p.m. Mon.-Thurs., 11 a.m.-2 p.m. and 6-11 p.m. Fri., 6-11 p.m. Sat., $9-15), with Southern country cooking like barbecue ribs and catfish.