While you start your question with the function of resistors, you are interested in a generic idea of how to learn the principles of circuit analyis in general.
The most common flaw of analogies is, that they aren't consistent at some point or the persons using the analogy fail to match every thing and aspect in the analogy and the real problem. The most dominant failure of the water analogy is, that the fluid resembling the electrons is not compressible. Electrons behave like an ideal gas in most media (vacuum and conductors), so if there's an analogy which fits best, it's a more or less pressurised gas flowing between parts of the circuits with differing pressure. But in general, forget phrases like "absorbs voltage". Because this wording does not fit the concept of voltage. It wouldn't fit to say, a small tube "absorbs the pressure" when using the water or gas model, either, would it?
But to get back to the basic things that happen in a circuit. If there's a current flowing in a circuit this must have a cause. In concentrated circuit elements (this is what we assume in basic circuit analysis, you don't have to mind about that now), the cause for a current to flow is, as already mentioned by @pjc50 an electrical field. The electric field represents the voltage between the two ends of a device with a defined geometric size regardless if it's a resistor, a diode or a spark gap.
What happens inside a resistor? Electrons have a mean free travelling distance, during which they are accelerated by the electrical field inside the resistor. After being accelerated over a distance and taking up kinetic energy, which may be more or less than this named distance, they accidentally and eventually hit some atoms (i.e. the orbitals of the non-free electrons), get deflected and loose their kinetic energy. The energy is now transferred to the atoms of the resistor resulting in vibrations (which is basically thermal energy). The resistor heats up.
When the electron leaves the resistor, it statistically has the same speed as it had, when it entered the resistor, because inside the resistor the acceleration due to the electric field and the deceleration due to collisions balance themselves! This is a very important property of ohmic resistance.
So, the whole topic is about balance, then. That's where algebra comes in. The speed of electrons traversing something like a resistor and the number of them doing that simultaneously at any cross section of the device are related to the voltage and the physical properties of the material. There are a lot of parameters and equations but they are fairly simple for an ohmic resistor. You should read about them and strive to solve them once for a resistor, then I think some basic understanding may arise about the concept of resistance.
After that it will be easy to understand the other circuit elements and finally getting an idea, what a circuit is after all.