Red Shell Homing Mechanics

Often when playing Mario Kart Wii, I'll have a Red Shell whizz right by me if it was just shot, or one of my Red Shells will just miss another player. Because of this, I'm curious how the Red Shell homes in on players - does it act as a green shell for a little bit before it decides who to home on?

Yes, just as in the 64 version, the shell will travel a short distance as if it was a Green Shell and then take off towards it's locked target.

This is also the reason why you have to be careful to shoot it in the direction of your target (i.e. not perpendicular) or else it has a higher chance to hit the wall in front of you before it veers off toward the guy it's meant for.

In addition to Pithlit's answer

Red shells do not using homing if fired backwards, they act like green shells, but do not bounce off objects.

Red shells cannot take sharp corners very well, so you can avoid them sometimes by taking a sharp corner