Detroit is a really great place to find an exciting variety of foods to suit your palate. There is Mexicantown, with legit Mexican food that will spoil you from ever eating at Taco Bell again. There is Greektown, which will spoil your senses with their desserts. The baklava is to die for in that area. There are also amazing treats that span from fine French dining to Korean/Japanese fusion. You can spend a lot of time in Detroit trying out all the different foods it has to offer, and it will be quite a while before you try them all. It's definitely hard to choose your favorite!
For most grocery stores, you will have to travel outside of Midtown. There's a Food Pride a couple blocks from campus and the Honey Bee Market in Mexicantown, but beyond that there's not really much. The shuttle on campus leaves weekly to go to the Meijer (a superstore like Super Walmart) in Allen Park for you to do grocery shopping. However, there are plenty of residents that have cars that are more than willing to carpool with you in case you're not free the time the shuttle goes, or if you just want to go somewhere else.