What exactly is meant when we say the universe is expanding? I understand stars and galaxies are getting farther and farther away from us, but are individual particles (atoms, molecules, anything found on Earth, etc.) drifting away from each other as well? If no, then why not? If yes, then shouldn't it, at some point, get to a point when particles can no longer bond with each other?
When we say the Universe is expanding, we really mean that space is expanding. The Universe could be infinite in size, but there continues to be more and more space between objects. Essentially, objects are moving away from each other because more and more space is being created between them.
In the early Universe, matter was much closer together than it is now; this density caused extreme temperatures and no hadronic matter could form. The first expansion is thought to have occurred because of how hot and energetic everything was, and spacetime itself was expanded in an event we call the Big Bang.
As objects spread out from each other during the inflationary epoch, the Universe began to cool down. Now, dark energy is the main culprit for its accelerating expansion.
To answer your question, yes, dark energy is making all particles move away from each other, although farther objects move away from each other at a faster rate. Why is that?
Well, the only thing holding all atoms, stars, planetary systems and galaxies together are the four fundamental interactions — the strong force, weak force, electromagnetic force, and gravity. In space, the main attracting interaction is gravity. As such, dark energy will cause farther objects (which are less affected by our gravity) to move away at a faster and faster rate.
Think of it like you are holding a dog with a leash, and the ground begins to spread out from below you. You and your dog are being moved away in opposite directions, but you both stay close together because of the strong leash.
Now, your mention of atoms being torn apart is actually the basis of the Big Rip hypothesis. It states that dark energy will become more and more abundant until even closeby objects separate from one another. First, galaxies will be disbanded as the stars move away from each other in all directions.
Much later, the gravity of planetary systems won't be enough to hold them together. Later still, planets and stars will be disbanded into their molecules. This will eventually continue as the amount of dark energy increases, until not even atoms can be held together by the electromagnetic force, and the subatomic particles will be moved apart.
By the time the expansion of the Universe exceeds lightspeed at the subatomic scale, no particle will ever be able to interact with one another. The Universe would dissolve into countless lonely particles that won't be able to do anything. Now, the Big Rip hypothesis is only one of the three most known fates of the Universe. Currently, the most accepted theory is the heat death of the Universe.