I had to read up on equipotential surfaces recently and I noticed the pattern of equipotential surfaces around a point charge. The idea came to me then that since no work is done in moving charges along these surfaces, perhaps electrons orbit around a nucleus along these regions?
You are probably thinking in terms of the shell model of the atom:
It is important to note that the shell model is ultimately inaccurate and is not really a good representation of how modern physics views atomic structure. It has a few advantages, and it is a nice tool to explain certain features in atomic structure (in particular, the fact that there are "boxes" of some kind that will only fit a certain number of electrons) but it is only that: a rough model that gets some features right but not others.
More to the point, it is incorrect to think of the electrons as sitting on a circular orbit and moving along it. It is much more accurate to think of electrons as occupying orbitals, which are cloud-shaped regions where the electron is more likely to be found than elsewhere.
In particular, the orbitals do not occupy strictly one radius, but they are spread out in space, so they do not occupy a single equipotential surface. This is perfectly fine: they will just have more kinetic energy in the regions closer to the nucleus where the potential energy is lower. (Note also that this is the case, too, for classical orbital motion, like the Earth's elliptical orbit around the Sun.) The ion performs work to speed the electron up as it comes closer, and to slow it down as it goes away; overall, energy is still conserved, but there's nothing stopping different parts of the system from exchanging energy over time.