"This is kind of broad, but what are some of the rookie "mistakes" that low-budget productions and student productions most often make, that you think they don't have to?"
In addition to David's advice concerning the camera...
BAD SOUND -- Sound is as important as image. Every location should be checked out for background noise. Get a GOOD sound guy with a good mic(s) and setup.
SHAKY CAM -- God I hate shaky cam. Anything more than a few seconds, say when an explosion goes off, is too much.
NOT ENOUGH COVERAGE -- This will kill you when it comes time to editing. You will suddenly have continuity errors that you hadn't expected.
BAD ACTING -- What more can I say about that?
BAD SCRIPT -- You either have a good script or you don't.
May the force be with you.