Keep an eye out at Furukawa Fish Market — you're likely to find a great travel trinket in one of its many stalls. After you've admired the wares, try some fare from one of the nearby food vendors.
When you've finished at this community marketplace in Aomori, why not make your way to Aomori Museum of Art, Nebuta Museum WARASSE, and Aomori City Forestry Museum? They're near Furukawa Fish Market and also a hit with many tourists.