The Richard D. Schwartz Observatory:
Run by the Department of Physics and Astronomy at UM- St. Louis, the observatory houses 16" Meade LX200 telescope and an SBIG ST-8 CCD. Public open houses are held February-November, with an astronomy student operating the telescope and showing objects of interest in the night sky. Also check out their Facebook page!
The Crow Observatory: Located on the Washington University Hilltop Campus, the Crow Observatory is open to the public Monday-Thursday evenings during the fall and spring semesters (weather permitting).
Astronomical Societies & Public Viewings
Astronomical Society of Eastern Missouri: A St. Charles County based astronomy club, they are involved in the community through various public outreach programs and weekly public viewing sessions.
The St. Louis Astronomical Society: The main St. Louis astronomy club provides resources for finding open viewing sessions, and their monthly meetings are open to the public free of charge.
Fun Astronomy Related Sites
National Aeronautics Space Administration (NASA): Stay up-to-date with current missions, learn about past missions, and read about future missions. Plus, this is a great resource site for general interest in space.
HubbleSite: Dedicated to the Hubble Space Telescope, this site provides lots of information on the telescope and what it has done. It also includes a full archive of the images taken by the HST (they make great desktop backgrounds or screensavers!).