Siegfried's Delicatessen (20 West 200 South 801/355-3891, www.siegfriedsdelicatessen.biz [1], 9 a.m.-6 p.m. Mon.-Fri., 9 a.m.-5 p.m. Sat.) has a great selection of sausages, cold cuts, breads, pastries, and cheeses. Another good place to stop for provisions is 
Tony Caputo Market and Deli (314 West 300 South, 801/531-8669, http://caputosdeli.com [2], 9 a.m.-7 p.m. Mon., 9 a.m.-9 p.m. Tues.-Sat., 10 a.m.-5 p.m. Sun.), with great sandwiches, Italian-style sausage and cheese, loads of olives and other Mediterranean temptations, and a park nearby where you can have an impromptu picnic. Carlucci's Bakery (314 West 300 South, 801/366-4484, www.carluccisbakery.com [3], 7 a.m.-7 p.m. Mon.-Fri., 8 a.m.-5 p.m. Sat.) offers European-style breads and pastries.
In summer, the Salt Lake Farmers Market (300 South 300 West, 8 a.m.-1 p.m. Sat. June-Oct.) takes over Pioneer Park. The market offers a wide variety of ethnic and to-go eating options, along with a selection of fresh fruits, vegetables, and baked goods.
Links:
[1] http://www.siegfriedsdelicatessen.biz
[2] http://caputosdeli.com
[3] http://www.carluccisbakery.com