How do magnets move towards or away from each other? (Explanation on a molecular level)

I don't quite understand your last sentence. In the case you're describing, the two magnets attract each other and therefore accelerate towards each other. Of course the poles of a macroscopic magnet influence each other on a molecular level, but they arrange themselves in a stable configuration which results in the macroscopic north and south poles of the magnet. Also the sides that are not facing each other influence the whole process, but they only contribute in diminishing the net force that is pulling the magnets together.

If you want to understand this better you should first look at how electric point charges interact with each other and then how this behavior translates to electric/magnetic dipoles.

An atom with an unpaired electron has a magnetic dipole moment. In a ferromagnetic material a significant fraction of the dipoles can be brought into alignment and will maintain that alignment in spite of thermal agitation. In a bar magnet, the alignment of the dipoles produces a magnetic field which is very similar to the field that would be produced by an electric current flowing around the surface of the magnet. The field leaves at the north (seeking) end of the bar, spreads and curves around to re-enter at the south (seeking) end (forming a closed loop). If the north pole of one bar is brought near the south pole of another, each atomic dipole tries to align with the resultant field and each dipole in the spreading field of the other magnet is subject to a force toward the other magnet. If one of the magnets is reversed, this force is reversed. You can think of each dipole as a very small current loop, or you can analyze this force in terms of the (imaginary) current loops flowing around the outside of each magnet. (In an external uniform magnetic field, there is no net force on a magnet. A compass needle rotates, but does not try to translate.)