For me myself one of the easiest concepts to understand was quantum annealing (https://en.wikipedia.org/wiki/Quantum_annealing). It does not mean that's the strongest point of quantum computers, but it's a nice way to start.
By the way - you could as well simulate a quantum computer on a classical computer; it'd just be incredibly slow.
It describes the kinds of computations that need reversible computations and kinds of devices that can perform them. That makes it easier to notice similarities, as quantum computers are not the only ones in this category. Notably, photonic and low power computations call for reversibility.
The first couple of pages of the paper also touch upon what can be done with them - while you don't have to throw away all you knew about logic gates, the reversible ones are their own thing.
If the water goes too deep, perhaps Mika Hirvensalo's "Quantum computing" helps. It approaches the topics of how to compute with qubits from mathematical side.
I still don't know the theory behind quantum annealing though - I don't think these two cover it.
Still, I learned that the main difference is that the state doesn't have to travel as with electronic circuits, but rather one would likely "apply" gates as "events" acting on the state. A qubit could then be any quantum object - a electron trapped in one place and being hit with photons for example. Place a couple together and hit them in a coordinated way and that event was your gate.
When it comes to saving states, it will be problematic. Of course, it's possible to "save" a qubit to another qubit - with some workarounds for the "no-cloning" principle.
But if you mean saving them to a classical system, it seems it's not possible at all. A qubit is a representation of a probability, and once read it's destroyed. You could try recalculating it several times and get an idea of the probability it represents. However, qubit readouts are not guaranteed to be independent from one another, giving another bit of headache here.
Disclaimer: I'm not a physicist - just a software engineer trying to understand the hype.
* https://www.youtube.com/watch?v=F8U1d2Hqark * https://www.youtube.com/watch?v=ZoT82NDpcvQ
Like, how does one produce logic gates out of them? How do you create an IEEE 754 Floating point multiplier? I get that they're supposed to represent overlaid states on the same device due to Pauli exclusion, but how do you separate out states and use them in logic? Or save states? What happens if logic produces the same state for one device? Is standard metal used to send signals, or is it something fancy-dancy like neutrinos?
It's honestly the most exotic thing I know of in tech, just trying to even begin to get a handle on it, never-mind trying to perform useful work out of them.