I see some kind of slug or larvae in my tank. They're just on the left and right side of the tank. They are very small, less than 1 mm and they travel fast. They have a white body and its very slim. I recently bought some red cherry shrimp, could this be their young? I also bought a new plant this weekend and some kind of organism might have traveled with it. Not sure though.

Sounds like a planaria or some other type of worm. Nothing to worry about, they are normal.

Probably planaria...come on..don't you remember middle school experiments with asexual reproduction (those folks in Knoxville, TN are familiar with that method--I hate Rocky Top)....you can cut planaria in half and they'll divide into two separate little buggers!!

I always had to cut them in pieces for regeneration experiments. My Bio teachers all seemed to be obsessed with 2 headed - 2 tailed planaria.

So that's what those things are, b/c I see them in my snail tank as well. They are everywhere and definitely move faster than the snails.

Do fish eat them? Are they harmful?

Some fish eat them, and they are completely harmless scavengers.