You might want to spend some time browsing the exhibits at Astrakhan City History Museum, Museum of Astrakhan Culture, or Astrakhan State United Historical and Architectural Museum Reserve. Sights like Velimir Khlebnikov House Museum and Museum of Military Glory 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.