I couldn't fit this in a comment, sorry. FirstlyFirst, let's get the basics straight.
Basics
There are only four fundamental forces of nature: gravitational, electromagnetic (EM), strong and weak. Energy is transferred by the mediators of the forces, which for the electromagnetic force is the photon. For more on this, check out Quantum Field Theory and the Standard Model.
Matter-Matter
A common misconception is that atoms physically touch, they do not. Instead they electrically repel each other. The other forces aren't as relevant here, because gravity is much weaker and the strong + weak forces operate at a much shorter range.
Matter-radiation interaction
EM absorption is really a form of interaction between fermions and electromagnetic radiation. Regular matter can absorb EM radiation in a variety of ways: electronically, translationally, rotationally and vibrationally.
The strength of the absorption depends on how similar the energy of the EM radition and the atoms' transition are (more similar means stronger absorption). Yes, I've left a lot out and if you want to learn more about this, get a good textbook on quantum mechanics.
Example - Hydrogen's electronic transitions
Credit for this excellent image goes to Szdori, and was taken from https://commons.wikimedia.org/wiki/File:A_hidrogen_szinkepei.jpg.
If your EM radiation has a wavelength of 122nm, then it is likely that an electron in the ground state of an hydrogen atom could absorb the photon. But it is very unlikely that the 103nm transition will absorb it, because the wavelengths (and thus energies) are different). The exact probability of a transition absorbing a given wavelenegth can be calculated with quantum mechanics (again, I'm leaving this out, see a textbook for the full details).
An important case to note, is that of ionization. This is where the electron transitions out of a bound state, and is freed from the atom. This can happen with any EM radiation with enoughsufficient energy, but get's less likely as the energy gap between ionization and EM radiation increases.
Penetration
EM radiation will penetrate matter, if it's sufficiently unlikely to be absorbed by the matter while passing through it. This will depend on the wavelength (or equivalently energy) and the wavelength of transitions present in the matter. If there are transitions of similar wavelength, then it's likely that the photon will be absorbed. Conversely, if there's no similar wavelength transitions, then the photon(s) probably won't be absorbed.
Radio waves penetrate most matter very well, because they're too low energy to excite any transitions.
Infrared EM radiation can be absorbed by the translational, rotational and vibrational transitions, hence Infrared radiation is associated with heat.
Visible light can be absorbed by the electronic transitions or common matterof everyday atoms, hence we can see it.
UV is likely to interact with some electronic transitions, and also the ionization transition (hence it's dangerous to be exposed to too much UV).
X-rays and Gamma rays penetrate normal matter very well, because they're so energetic that the only transition that they're likely to interact with is the ionization transition. The interaction is still unlikely, because there is still a large gap in energy.