The block around the Birmingham Civil Rights Institute on 6th Avenue North has become a de facto museum district. Within 200 meters of each other you'll find the Civil Rights Institute, the 16th Street Baptist Church, Kelly Ingram Park, the Carver Theatre (now the Alabama Jazz Hall of Fame), and the Birmingham Public Library — all anchored by Linn Park to the north and the Civil Rights District to the south.
Plan this as a full morning or afternoon: start at the Civil Rights Institute (buy the ticket, take your time), walk through Kelly Ingram Park, visit the church, browse the Jazz Hall of Fame, and end at the Birmingham Museum of Art if you still have energy. This compact area contains more American history per square block than almost anywhere outside of Washington D.C.