When photons reach us, what exactly is happening to us and to that photon?
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 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
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.