Ceiling fan replacement, one wire is always hot
This looks super simple.  I mean normally, wire colors are a mess and don't really correspond to wire function.  But luckily, here, they correspond exactly: 


White = Neutral
Black = always hot
Red = switched hot


So the switch gets black and red becuase its job is switching.  The lamp gets red and white because you don't want it on 24x7.  

You join all blacks together, all whites together, all reds together (pretending the hot lead from the lamp is a red).  And you're done.

We don't know how it was hooked up before, so I can't guess why this didn't work before.  I have a feeling the old lamp was wired into always-hot (black) or the switch was miswired. 

Why does the red report hot when the black is? Because I think your non-contact tester is picking up on "phantom voltage", which is a bit of capacitive coupling that can happen between wires in the same cable.  Wires left to float can do that.