How to I get linux to stop prompting me for a password and remember it? For instance, when using the root user, I always get prompted, but I would rather only be prompted once per user session.
In terms of the console (bash) there is no way to do that. You could put yourself in the "wheel group" and you would only have to type
sudo xyz instead of
su [Enter] xyz and it won't bug you every time. In terms of the GUI (I see you tagged Ubuntu) certain tasks (those found in the System -> Administration menu in GNOME) will remember the root credentials for a set time or until you log out, automatically.