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