You might want to spend some time browsing the exhibits at Astrakhan City History Museum, Astrakhan State United Historical and Architectural Museum Reserve, or Museum of Astrakhan Culture. Sights like Museum of Military Glory and Velimir Khlebnikov House Museum highlight some of the local culture. You can watch performances at Astrakhan State Theatre for Young Spectators, Astrakhan State Opera and Ballet Theatre, and Astrakhan Theatre For Young Spectators if you're interested in local theater.