How do EM waves cause molecules to rotate, move, and cause electrons to move up energy levels?
But how do EM waves cause atoms to rotate, and vibrate?


In no way do EM waves cause atoms to rotate and vibrate. Atoms (more precisely, nuclei) in a molecule are always moving, so EM radiation can only change the way they move.

The thing is that within the Born–Oppenheimer approximation an energy  state of a molecule is the sum of the electronic, vibrational, and rotational components,
$$
  E = E_{\rm{el}} + E_{\rm{vib}} + E_{\rm{rot}} \, ,
$$
and there are accordingly three types of states and energy levels: electronic, vibrational, and rotational. For each and every electronic state there are few vibrational states associated with it and for each and every vibrational state there are few rotational states associated with it.

A transition can happen not only between different electronic states, but also between different vibrational states associated with the same electronic state or between different rotational states associated with the same vibrational state. Different vibrational and rotational states correspond to different ways atoms of a molecule move in space, so vibrational and rotational transitions change the way atoms move.

And, of course, any transition emits or absorbs a photon, the frequency of which is proportional to the difference in energy levels. Vibrational and rotational transitions spectra are studied by rotational–vibrational spectroscopy.