For instance, in the first Bohr orbit a classical electron (particle) in hydrogen would move in a circle of radius 0.53 angstroms (5.3 x 10^-11 m) at a speed of c/137 - a bit less than 1% of the speed of light. This tells us that we can model the ground state of hydrogen reasonably well as a first approximation ignoring special relativity and introduce it later as a perturbation.
As an atomic physicist, I like to joke that we understand just one atom, hydrogen, mainly because it's a simple system with exact solutions, and we can build a lot of intuitions starting there. So you want to extend to think about carbon. First, because carbon (ordinarily) is electrically neutral, there are as many electrons as protons - a total of six. Imagine taking all the electrons away and building the neutral carbon atom starting with a nucleus, then adding electrons one by one.
The first electron sees the bare nucleus - the chief differences from the hydrogen case are that the nucleus is more massive and has more charge (six times more, as there are 6 protons). Classically, we can model this ground state just like in hydrogen, but now the electron will be closer to the nucleus and will be moving faster (both thanks to the higher nuclear charge). Relativistic effects matter more but not all that much more; we can apply the same nonrelativistic quantum mechanics as we could for hydrogen and we end up finding that the lowest energy state has zero angular momentum (an "s" orbital).
Something similar happens when you add the 2nd electron - it will also end up in an s orbital. The energy will change a bit because the electrons repel one another electrostatically, but basically these first two electrons are in the same state except for one important detail: if you measure the orientation of the internal angular momentum of one ("spin" ), the other must have it directed in the opposite direction. In other words, their spins must be opposite, and this is because electrons are "fermions" - antisocial particles that refuse to share the exact same quantum state as other particles of their kind. The application in chemistry is called the Pauli exclusion principle.
As a side note, spin is a pretty mysterious yet fundamental property. I recall reading that Richard Feynman gave some criterion for understanding a concept that included being able to explain it with resort to advanced mathematics, and when challenged to explain spin, concluded something like, "I have to admit I don't understand spin."
Anyway, it turns out that for "spin-1/2" fermions like electrons, there are only ever two possible outcomes for a measurement of spin orientation (usually just called "up" and "down" ). So if we bring in a third electron and try to stuff it into the ground state, there is no way to do so without violating the exclusion principle. Quantum mechanics tells us the next lowest energy state also has zero angular momentum (which we'll now call "orbital" to distinguish this from the angular momentum of spin), but sits a bit farther out. We call the state the first two electrons are in "1s" orbitals and this new one "2s," where the numeral is the "principal quantum number" and which you can think of as roughly relating to the size of the orbital. You can stick a second electron (with opposite spin) into 2s for a total of 4 electrons in s orbitals.
It turns out that the next lowest energy state has the smallest possible nonzero orbital angular momentum. Those are called "p" orbitals, and we can stick our last two electrons in those. It also turns out that we can stuff up to 6 electrons in a given p orbital, because there are three possible outcomes for measurements of angular momentum along a given direction given the amount of angular momentum in a p orbital (we say it has 1 unit of angular momentum, and if I pick a direction for measurement, the possible outcomes are along that direction, perpendicular, or opposite - typically assigned values of +1, 0 and -1). For each of these, we independently have two possible measurements of spin, so there are 6 (3 orbital x 2 spin) ways to assign unique sets of quantum numbers to electrons in a p orbital.
To do speed "right" what you would want to do is apply to the mathematical representation of a given electron's orbital (the "wavefunction" ) a mathematical "operator" that represents speed. This yields something called an "expectation value," which is the most obvious stand-in for what you're looking to understand. Because an electron is equally likely to be moving left or right (or up vs. down, etc.) the average *velocity* will be zero; but the expected kinetic energy (and thus speed) will be nonzero. And the results will in general be different for the different electrons, but will be roughly on the order of 1% of the speed of light. The "inner" electrons will be faster, the outer electrons will be slower.
Similarly, because the electrons are not things one ought to regard as having definite positions, defining a distance to the nucleus is a tricky exercise. But there are still useful quantities to be had. The first thing to bear in mind is that electron orbitals are much larger than nuclei (whose sizes are typically of the order 10^-15 m, or 1/10000 of a typical electron orbital in effective radius). Because there is a nonzero probability of an s electron being found "exactly" at the location of the nucleus, one could argue that sometimes there is zero distance to the nucleus! (And this fact results in some interesting correction terms that arise in the energies and wavefunctions for s electrons.) But similar to speed, we can calculate an "expectation value" for a measurement of the electron's location relative to the nucleus, and the results will show the trends I've already described, and have values appropriately measured in units like angstroms or maybe picometers (trillionths of a meter) for inner electrons (and larger - sometimes very much larger! - for outer shell electrons).
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