There are no shortage of southern and barbecue-style restaurants around Memphis. I have a few random ones to throw out that do not involve those food types, since others will probably give more input there.
Given that this is a relatively big city and very food-oriented, you could spend a week eating here non-stop just trying to cover all the good places. You have a lot of picking and choosing to do!
Sushi - Bluefin (downtown on Main Street). It's relatively expensive, but if you go for lunch, they offer half price sushi rolls based off chef's selection that you pick from a tray as they are prepared. It's still hard to stay under $15 per person, because you always want to try one of their high-end (more expensive) rolls. Very good stuff. Best sushi in town, I think.
Italian - Bruno's (in Bartlett, a large suburb on the NE corner of Memphis), is a newer place that I stumbled upon last year, and I've been very impressed. It's in an old barbecue restaurant, and is one of those places where you can get great food without waiting long for a table since it hasn't gotten popular yet.
Italian - PaPa Pia's (downtown, on Union Avenue. Closed for lunch as of this year). They are mostly known for their flatbreads, but I generally recommend this place for high quality food without high prices. Their Barbecue Chicken Alfredo Pizza is the last thing I ate there - weird, but tasty.
Chinese - Chang's House (also in Bartlett) is a long-standing establishment in a tiny strip mall one block off the main road through town. Order the chicken fried rice, spring rolls, and anything else to go with it. Expect to have leftovers due to large portion sizes.
American (high-end) - Flight (downtown on Monroe & Main Street). Lunch is affordable ($10-15), dinner is fairly high ($25-35). They offer "flights" of meals, which means you get three smaller servings of food that follow a single theme, instead of a single entree. They have really cool menu items, including fresh seafood, elk, buffalo, duck, and some really tasty Cajun dishes. Perfect for someone who can't make up their mind about what they want to order. Romantic atmosphere for dinner, unless you are traveling with dudes...
Cajun - Deja Vu (just south of downtown, in the hood). A hole-in-the-wall lunch establishment. Get there before 11:30 or you might not get a table. Order the etoufee if you want something tasty and reliable. Check out all the pictures of the celebrities on the wall for a smile.