What happens at the subatomic level when energy is added to an atom? For the sake of simplicity I shall use the hydrogen atom. I know that adding energy to the hydrogen atom means bombarding it with photons. I am curious about what happens to the quarks and electron of the atom. Does the electron move to ever higher electron shells as energy is added? Does the velocity of the electron increase? What happens to the quarks as they continue to gain energy?
 A: There are many was to increase the energy of a system. 
If you're meaning by form of electromagnetic radiation then the nucleus remains effectively unchanged. The electron can theoretically increase to energy levels infinite in distance from the nucleus however most of the time they disconnect from the atom due external forces. The electrons don't exactly move as such. They are actually distributed around the atom so its distribution around the atom would be across a much larger space.
One may also move an atom increasing its kinetic energy. From there we see all sorts of odd effects of special relativity. If I were moving at the same rate as the atom (or if in fact I were the atom) I would see no  change in its mass. However if I were still the mass of the atom would appear to increase so by logic the mass of every component would appear to increase. [I can't properly explain this in great detail.]
I could also split the atom by firing a neutron at high speed into the nucleus. In this case we see the weirdest things with the quarks as a top quark of a proton converts to a bottom quark to make the proton a neutron and vice-versa. This happens as the quarks exchange a charged boson. A charge boson could be said to 'carry' a third of the charge of the electron and so swap a two third up to a one third down. This is called quark transmutation by the electro-weak force. The same happens for fusion.
On another note you can't split quarks apart. The energy to split apart a quark pair is another to create another two quarks so that they will always be in couples. Romantic aye?
A: You say:

I know that adding energy to the hydrogen atom means bombarding it with photons. I am curious about what happens to the quarks and electron of the atom. Does the electron move to ever higher electron shells as energy is added? 

The energy of a photon to lift the electron to a higher orbital has to be quantized, i.e, cover the energy difference of the two energy levels(, within the  width of the energy level). It is not increased incrementally but in quantized steps, defined by the hydrogen atom solutions. The electron can move in these steps and finally reach the band where it can be essentially free of the attraction of the potential, and be detached from the nucleus having reached the ionization level ( the nucleus becomes an ion).

Does the velocity of the electron increase?

One could take the energy of the energy level and compute a velocity for the electron, but it has no meaning, as the electron is in an orbital, not an orbit, i.e. a locus where it is probable to be found when one scatters it to the ionization level. 

What happens to the quarks as they continue to gain energy?

The quarks are tightly bound within the proton of the hydrogen atom. The dimensions are such that the effect of the 1/r potential from the interaction of an individual quark with an electron compared to the strong interactions that keep them bound is tiny and not measurable anyway, since quarks are never free and appear only as "quark jets" in high energy interactions.
