Classical electromagnetic theory could not explain the spectral lines seen in light spectra. It predicted that an electron caught around a proton would radiate continuously (due to the radial acceleration) and fall on the proton by emitting a continuous spectrum of light. No stable hydrogen atoms would exist.
Here is the hydrogen atom:
The fixed orbits explained the observed spectra, the Balmer and Lyman series and this was one of the main pillars for inventing quantum mechanics.
Notice that the lines have a width
If when you shine a photon into an atom for example, and this excites an electron to a higher energy level,
Let us take the hydrogen atom as an example.
Yes, the atom will be excited if the photon has the frequency/energy of the difference in the energy levels
do the electron(s) keep going higher the more light you shine, and is there an energy limit, if so why?
It is not a matter of quantity but of the appropriate energy differences
Look at the energy levels . For a photon to hit an electron and ionize the atom it has to have a frequency h*nu=13.6eV. To transit to an intermediate level, the photon has to have the difference in energies, otherwise it just scatters off the field of the atom and leaves it intact.
If you give a photon the correct steps in energy , then the electron can step up until ionization. Again, it is the energy or the photon that has to match the energy level differences.
If you have a hydrogen gas, and a source with the appropriate energy levels to excite the atom then the more photons, the more excited states. General frequencies will ony by chance fit the correct difference.
Secondly, if you then stop shining light, why will the electrons fall back to a lower level?
There is a calculable quantum mechanical probability for the electrons to fall to an empty lower level, because it is a law of nature, quantum mechanical and classical, that everything goes to the lowest allowed energy level where stability exists.
Will they at all? And why? it seems arbitrary that they will unless acted on by something else.
The electron can either cascade down the levels releasing photons with the appropriate frequency/energy, or go in one step to the lowest energy level. All these are calculable probabilities in the quantum mechanical frame. Your "acted on by something else" translates in saying that the Schrodinger solutions are not enough to give widths to the predicted lines, which is true, one needs quantum electrodynamics.
A single atom in space, if in an excited state has a calculable probability to fall to the ground state ,which is modeled by using interactions with the QED vacuum.
If they do fall back, how long does it take till they do?
When you study further you will see that this is connected with the width of the line, which is not a strict energy line but has a Δ(E) again calculable. These calculations need quantum field theory , not the simple Schrodinger model, as valerio92 quotes in his answer. The Heisenberg uncertainty principle then ties the decay times with this energy width.