The North Carolina Museum of History, free and a short walk from the Capitol, covers the state's story from its indigenous cultures through the present day. The permanent exhibits are well-designed and genuinely engaging: the Civil War galleries are thorough and appropriately complex, the textile and tobacco industry exhibits explain how this state's economy worked for a century, and the civil rights section covers the Greensboro sit-ins and other local events with first-person accounts.
The museum also holds a significant collection related to the Wright Brothers' first flight at Kitty Hawk — including artefacts and documentation — and an exhibit on North Carolina's role in the American Revolution that is more interesting than it sounds. Budget two hours and you will still miss things.