Is that really a hydrogen atom? Is a hydrogen atom just an electron orbiting around a proton?

Well, long ago, physicists like Ernest Rutherford thought that might be a good way to picture a hydrogen atom, but, as we'll see later, the discoveries of quantum mechanics completely changed our thinking and our models of atoms.

So, does that mean that electrons don't go in orbits around atoms?

It's not the best model for an atom, but just recently, physicists figured out how to make a large number of electrons stick to particles of dust, and then made such charged dust particles go around a positively charged metal ball in the laboratory. Here's a movie of two charged dust particles in orbit around a ball (the orbit is seen from the side).

This movie's size is 364 Kb.