I’ll second the recommendations for the civil rights and Barber museums. Barber especially is a truly world class facility. If you’re a motor sports fan, and motorcycles in particular, it’s a must visit. I know nothing about motorcycles but I still go there every few years and am amazed at the collection. CSB - I was on the design team for the museum expansion that was completed a few years ago.
The civil rights museum is definitely worthwhile, but it’s a somber experience that examines some pivotal moments from the civil rights era, some of which occurred in the surrounding blocks. After you’ve toured the museum walk across the street to Kelly Ingram Park and the 16th Street Baptist Church to see the actual locations of some of the events depicted in the museum.
The restaurant scene in Birmingham is solid, and the restaurants mentioned up thread are all good recommendations. Highlands Bar & Grill is great and was the national Beard award winner a few years ago. Chez Fon Fon is next door and is owned by the same restauranteur. It’s also quite good and not as expensive.