How do we know not all photons are absorbed? Only those of specific energies? When a photon hits an electron in an atom, its energy has to be equal to the difference in energy between the current shell and a shell with a higher energy level, otherwise it is not absorbed at all. 
How do we know not all photons are absorbed? Wouldn't at least some energy of the photon be absorbed since it is an oscillation in the EM field?
 A: 
How do we know not all photons are absorbed? 

You can shine monochromatic light on a sample of a material and see whether it is absorbed or not.
For example if you shine different wavelengths of light through a chamber containing a pure gas, and you'll see that only wavelengths that match transitions of that gas molecule are absorbed. Wavelengths that are either too long or too short are not absorbed.

Wouldn't at least some energy of the photon be absorbed since it is an oscillation in the EM field?

It's not possible to absorb "some" of the energy of a photon. The whole concept of a photon is that it is the quantum of energy in the EM field. That means that when something exchanges energy with the EM field, it does so in units of whole photon energies, not fractions of the photon energy. 
A: Here is the spectrum of light coming from the sun.


Solar spectrum with Fraunhofer lines as it appears visually.

The  spectrum seen has all the wavelengths ( frequency = c/wavelength), and the dark lines are the absorption lines. Absorption and emission lines are one of the reasons quantum mechanics had to be invented.

Wouldn't at least some energy of the photon be absorbed since it is an oscillation in the EM field

The photon is a point  elementary particle of energy=to $hν$, it is not a classcical electromagnetic oscillation. It is described by a wavefunction which is not measurable. The complex conjugate square of the wavenctions give the probability of finding the photon  at (x, y,z,t)  with energy E , and the dark lines show an overwhelming  probability for the specific frequency/energy photon to be absorbed.
Light is composed by zillions of photons superimposed,  but a photon is not light. (a buildingis made of bricks, but a brick is not a building is a classical analogue)
A: We can shoot photons of different energies at atoms and see what goes thru and what is absorbed.  only the specific energy photons will be absorbed. And no part of the energy will not. This is one of the results from quantum mechanics. The first excitation energy of hydrogen is 10.2 ev.  If you shoot photons of energy 12 ev, part of it will not excite the H atom. You need a photon of 10.2 ev to excite the H atom. 
