At the smallest level, how do things move? When we see something moving on a screen it's usually just pixels being turned off at one location and turned on at another.  For example:
This would render a dot moving from A to C.
Turn on pixel A,
Turn off pixel A,
Turn on pixel B,
Turn off pixel B,
Turn on pixel C
In a real world scenario - how do things actually move?  How does the tiniest physical space get covered?
There must be a tiniest physical space right?  If not then from my reckoning we shouldn't move at all.  To move 1 step forward I'd have to first move half a step, but before that I'd have to move a quarter of a step, before that an eighth, etc.  I would always be moving forward but I would never reach 1 full step.
So assuming there is a tiniest physical space, does matter get turned off at point A and on at point B etc?
 A: Your question is still somewhat a matter of philosophy and the particular interpretation of quantum mechanics you follow because we can't yet probe probe to the tiny distances needed to truly understand this.
You question is somewhat similar to Zeno's paradoxes which can be solved mathematically via calculus.  Unfortunately calculus assumes everything is smooth and space is infinitely dividable which is at odd with quantum mechanics.
Some points about you question do have answers though.
Regarding shortest distances / quantization of space:
There isn't a tiniest physical space in the classical notion of what that means.  Space is not a bunch of tiny "pixels" (or "voxels") but there is the Planck length and the meaning of length below this is somewhat undefined.
Regarding something "turning off at A" and "turning on at B":
Remember particles (really, anything) can't be localized to an exact position due to uncertainty principle.  It's wrong to think of something occupying a point and when it moves, occupying the next point in space.  Particles are defined by a wave function which is a probability distribution and always has some physical spread to it.  You can think of motion at extremely short scales as the peaks of the probability wave function shifting along the wave in space.  If you think about the wave function as it relates to quantum tunneling you'll see that thinking of motion in the classic sense doesn't work.  If it did work that way particles would not be able to tunnel.
A: What you describe, "move 1 step forward I'd have to first move half a step, but before that I'd have to move a quarter of a step, before that an eighth, etc." is a paradox but only when time is continuous. If the time is also discrete, you jump from location to location instantly. You certainly can think of this as: "turning off at A" and "turning on at B" after a fixed period of time. So, there is no reason to worry about Zeno's paradoxes.
One can imagine that the space is "quantized"; how about a graph? It might be a grid or a random "mesh". A moving object (such as a particle) then occupies a single node of the graph at every moment of time. Is it possible to incorporate the uncertainty principle so that it can't be localized to an exact position? Yes, the particle is defined by a wave function which is a probability distribution spread over several nodes of the graph.
