Advice removing water stains on front curtains

Take them down and go to a dry cleaner and let them deal with the stains.
Almost anything else that you can do will involve a laundry and various household products with no guaranty as to the results. The stains are probably just dirt and dust that had settled on the curtains through the years that once they got wet, just showed up on the curtains. They should come out easily but the age and condition of the fabric may cause a problem.