“What happens at the subatomic level when energy is added to an atom?”
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.