I grew up in Wilmington (lived in NYC, DC, and Philly) and returned here about 10 years ago. The previous message about the "Pit" (Charcoal Pit) has been true for at least 35 years. One of the great hamburger joints on the East Coast.
For a great hot dog, fries and a root beer, The Dog House cannot be beat anywhere. Do not be put off by the looks of the place. It is found on Route 13 about 5 miles south of Wilmington near New Castle. They have foot-long hot dogs grilled in front of you. I usually keep it simple with chopped onions and mustard.
China Royal is hard to find north of Wilmington, but serves chinese food which is excellent. Any of the menu items are excellent, but I usually order chicken with pine nuts.
For lunch, The House of Saigon in Newark is highly recommended. Choose from one of the pre-set lunches in the back of the menu. Chicken with bean curd is superb.
Brunch in the Green Room at the Hotel DuPont is served in classic style. For dinner, you might try the Brandywine Room which is filled with Andrew Wyeth paintings.
We have sphagetti houses which are too numerous to count. Our Little Italy, on Union Street has three or four which have not changed in 40 years. Try Mrs. Robino's, Attilio's, or Pala's.
"Hoagies" are a Philadelphia area item, which do not seem to travel outside this region. Wilmington has a rivalry for the best and Casapulla's and Capriotti's duke it out each year for the best. Make sure you get an Italian Hoagie (aka Submarine sandwich) with crushed hot peppers.