Information on photographic noise and flash?

Help! I keep getting noise in my photographs. I realize having low ISO will help, but I think having the flash on all the time contributes to the noise and also I don't want "washed out" photos. I know a simple explanation would be to shut the flash off, but unless I have really good light (like when I'm outside) I end up with really dark pictures. If you have any ideas please help. My camera is a Kodak Easyshare Z712 IS.