It took years of research to come up with my own little list of places I couldn't live without.
1) Thurns on Greenlawn Ave - they're only open to the public three days a week (Thurs, Fri, Sat), but they're my go to spot for anything smoked or in a sausage. They are a good old fashioned charcutier that has been around for over 100 years.
2) Carfagna's Market on E. Dublin-Granville Rd - a true Italian grocery, with all manner of prepared foods, imported ingredients, an impressive butcher counter, and fresh local veggies in the summer. Decent wine and cheese selection.
3) Mr. Meatball on Cleveland Ave. in North Linden - a shop where they make the most amazing ravioli and meatballs
4) Schuman's on Harrisburg Pike on the West Side - butcher shop/deli with great prices and an incredible selection.
5) Trader Joe's (Dublin or Easton) - not sure if where you are moving from had a Trader Joe's, but they've got tons of staples (nuts, oils, pasta, cheeses) at really good prices.
6) +1 on the Jeni's recommendation, but if you get a chance, don't pass up Handel's ice cream either. Incredibly good, lots of flavors, Ohio company (out of Youngstown).
So many more recommendations - is there any certain part of town that you're focusing on?