When photons reach us, what exactly is happening to us and to that photon? I'm new to physics and am just going through some of the free online classes at World Science U, and after watching this video on the nature of the speed of light and its constancy, a question came to mind about photons.  (Video:  YouTube Video, World Science U course)  
I know that photons don't have mass, but what happens when photons — even the photons from distant stars — reach us?  Are we merely observing the occurrence of photons moving through space relative to us, or are we really being "bathed" in photons?  I know that when I observe rain, I can both observe it from a distance but could also be immersed in it as well if in the path of that rain.  But with distant starlight, are we just observing it or are the photons actually reaching and penetrating the earth around us?  If they are penetrating, does science tell us what is actually happening on an atomic or sub-atomic level?
 A: A photon is emitted by a star, travels in a straight line* and enters your eye when you look at that star. You see the star continuously shining because photons are continuously being emitted, so you can't actually tell when each photon enters your eye. It's like a constant flow of water as opposed to dripping.
You can't see photons in the same way you can see rain from afar. You can only effectively feel the rain when it hits your skin, i.e. you can only see stuff when the photons emitted or reflected from that stuff enter your eye.
*Strictly speaking gravity can bend the path of light, but that's general relativity
A: 
I know that when I observe rain, I can both observe it from a distance but could also be immersed in it as well if in the path of that rain. But with distant starlight, are we just observing it or are the photons actually reaching and penetrating the earth around us?

When you're immersed in rain, you interact with it directly: you get wet. That's similar to what happens when we see light.
When you observe rain from a distance, you interact with it indirectly: you don't get wet. Instead, you're interacting with light which has previously interacted with the rain. If there were no light (or sound, etc.), you wouldn't know the rain is there until you immerse yourself.
So can we interact indirectly with light? It turns out that we can't. Light waves don't interact with each other; they pass straight through each other unchanged, for example if we shine one beam of light through another beam (although they do interfere with each other in the place where they meet!). In terms of particles, we say that photons are bosons. Compare this to, for example, spraying one stream of water through another: they'll crash together and the water will fly off in a different direction (we say the water particles are fermions).
I suppose I should mention that, when the light has enough energy, it is possible to get an interaction: known as pair production, but this is a rather unusual consequence of relativity and quantum mechanics.
A: Yes, the photons actually reach you, like rain falling on you, not like watching rain from a distance.  When you see a star, photons from the star actually enter your eye.  
In for example rods of your eye, the photon causes a molecule of retinal to react by change from cis to trans isomer.
A: 
I know that photons don't have mass, but what happens when photons —
  even the photons from distant stars — reach us?

Small portion of photons is absorbed by atoms which are part of molecules which are part of retina. The energy of these absorbed photons is transformed to electromagnetic impulse. The produced impulses reach brain unit responsible for image recognition. After it recognized, for example, the star on the sky, the brain (mind) reacts correspondingly.
In your case, when photon reached your eye, you asked stackexchange which caused thousand of views :)
Some other photons are absorbed by body and their energy is transformed to heat (in case is star is Sun).
And lots of photons is dissipated and reflected by body. It means during daylight (star=Sun) or in the deep night (with no electric light) all you see is a reflected light from stars (including moonlight).

Are we merely observing the occurrence of photons moving through space
  relative to us, or are we really being "bathed" in photons?

We are really being "bathed" in photons. We are being bathed in visible and invisible spectrum of photons. 

I know that when I observe rain, I can both observe it from a distance
  but could also be immersed in it as well if in the path of that rain.
  But with distant starlight, are we just observing it or are the
  photons actually reaching and penetrating the earth around us?

They are reaching and penetrating us and the world around us. Even you see object near you your eyes and brain react when photons reach you and the brain's image recognition unit creates picture for you and illusion that the object is 'there'. So in rain case there are two thing: rain itself (the other story) and light which corresponds to the rain visibility. In latter case it is just star (Sun) light reflected and refracted by water drops.

If they are penetrating, does science tell us what is actually
  happening on an atomic or sub-atomic level?

Science tells us the one kind of energy is transformed to another one. Light as electromagnetic kind of energy if transformed to heat when it is absorbed. Or it can be reflected or refracted. On atomic level, the atom changes its energy level to higher or lower if it takes or emits photon. When photon is absorbed it magically disappears and atom becomes hotter...
A: 
If they are penetrating, does science tell us what is actually
  happening on an atomic or sub-atomic level?

You only have or see "photons" when the interaction process between the electromagnetic field (light) and matter (you) breaks spontaneously the time-reversal symmetry. This can be expressed in different ways, like saying that the vacuum state in not preserved by the interaction. The localization of energy in matter only arises because the interaction process is not reversible. This can be due for instance when a discrete atomic level is coupled to a continuum of levels (in the above molecular example). The time evolution will be given by Fermi-golden rule for instance, the time evolution is linear (with a constant rate), the process is irreversible and energy from the light field is being built and localized within the atom. A highly destructive process which does not conserve the energy in the field. This is different from the interaction with two identical discrete energy levels where the interaction is reversible for instance (Rabi oscillations) and as long nothing "perturbates" the process you will not see photons but oscillations of the field instead (the wave nature if you want). 
A: Not an real answer, since I have not physics degree too, just a modest addendum:
DavePhD answered:

In for example rods of your eye, the photon causes a molecule of retinal to
  react by change from cis to trans isomer.


While one of light role in nature is to carry information (direct - from source / indirect - reemission), then imagine this:
If you were looking at a clock ticking and then you started to move back at the speed of light the clock would appear to freeze because no new photons would reach you. 
If you were looking at a clock from distance of 1 light-year unit and then you move to the clock with the speed of light, then clock would tick forward through whole year in no time.
