Fairbanks [1] lacks a downtown grocery store, but you’ll find Safeways at University Center on Airport Way and in the Bentley Mall on College Road, Fred Meyer shopping centers at 3755 Airport Way and on the corner of Johansen Expressway and Old Steese Highway, plus a Wal-Mart Supercenter on Johansen Expressway.
Out on College Road at Caribou Way, the 
Tanana Valley Farmers Market (907/456-3276, www.tvfmarket.com [2]) is the place to buy locally grown produce, flowers, baked goods, arts, and crafts. The market is open Wednesday 11 a.m.–4 p.m. and Saturday 9 a.m.–4 p.m. mid-May–late September. This is a wonderful place to get a taste of Alaska (literally).
Links:
[1] http://www.moon.com/destinations/alaska/the-interior/fairbanks
[2] http://www.tvfmarket.com