Other than electrons, does light interact with the other subatomic particles? Also, do different elementary particles behave differently when interacting with light (X-rays or gammas)? Can you say, for example, that this is an up quark and this is charm?
Light, at the level of particles which you are asking, is composed out of a confluence of photons. Photons are elementary particles in the standard model of particle physics. These particles are at the underlying level of all matter as we presently know it.
To first order, photons interact with the electromagnetic interaction with all charged particles in the table, and thus with all complex particles , atomic and molecular entities that contain them. For example the neutron is neutral, but the quarks contain in it are charged thus there is a first order interaction.
Neutral elementary particles, other than the photon, are the neutrino and the Higgs. There is no first order interaction for these neutral elementary particles.
But interactions of elementary particles go through higher order terms in the expansion for calculating the crossections. There exists photon-photon scattering whose crossection grows with energy. There also exists neutrino photon scattering through Z and W exchanges in higher order diagrams, this reaction is important for cosmological models.
Equivalently there exists a higgs boson to two photon decay, so in cosmological studies the higher order diagrams would be again important.
So the answer to your question is:
Light interacts to first order ( higher crossections) with all charged elementary particles and through higher order diagrams with the neutral elementary particles.
All elementary particles with an electric charge interact with light. Uncharged particles don't interact with light.
Light, including everything from radio to gamma rays, only interacts with particles that have an electric charge. This is because light is the fluctuation of the electromagnetic field.
The most common result of interacting with a photon of light is a change of momentum (Compton scattering), so, all else being equal, there's little difference to what happens when a single particle is struck by a photon. The mass of the struck particle would affect the energy of the scattered photon, with larger mass particles taking less energy from the photon.
What determines more complex behavior (absorption, pair production, etc.) is the larger system to which the struck particle is bound--quarks in a hadron, electrons in an atom, etc.