I have an HP desktop and whenever I turn it on an "HP Advisor" bar appears at the top of the screen. I can click the "x" and it shuttles to the information bar at the very bottom of the screen but is there any way I can prevent it from appearing in the first place?

Hello.
Click START then RUN
Type MSCONFIG and press enter.
This will show you a list of the programs that auto run on startup (if the box on the left is checked)
Uncheck the HP box.
Exit and restart your computer.
(this is a good time to uncheck anything else you don't need to start on startup , this speeds up your PC startup time)

I know this sounds really dumb but I can´t find "run" on my Vista Classic control panel. What caragory would it be under?

You can use the shortcut , press the windows key(the one to the left of the left ALT key) and R at the same time.