So I'm playing through Skyrim and just now realized I can install the game to my hard drive - if I do so, will I lose my saved games?
No, you don't lose anything by installing an Xbox 360 game to your hard drive. (Well, unless you delete the saves to make room for the install... so don't do that...)
Load times are decreased slightly, but otherwise the game plays exactly the same. Note that you'll still need the disc in the drive in order to play.