Just a couple of quick comments...
If you were taking a class in electronics the 1st thing you learn is Ohm's Law which defines the relationship between voltage, resistance, and current (Volts, Ohms, and Amps). In the 1st semester you'd learn about how all of this works in series and parallel circuits and you might learn about using capacitors and inductors in DC circuits. Capacitors and inductors are more "interesting" in AC circuits but that would probably wait 'till the next semester.
There is a "standard water analogy" where water pressure is voltage, water-flow is current, a skinny pipe is high resistance and a fat pipe is low resistance. (Electrical resistance is the resistance to current-flow.) The main difference is, if you cut a pipe you get zero resistance and water flows-out all over the place. If you cut a wire you get infinite resistance and no current flows. Also, nothing bad happens with zero water resistance (except maybe a flood. ) but zero electrical resistance is a "short circuit", you get excess current and you can burn-up a power supply or blow a circuit breaker, etc.
The "trick" is knowing when the value is critical and when it's not. That only comes with learning and experience. Most of the time the value is not critical or "bigger is better". Sometimes the type of capacitor matters... Electrolytic capacitors (most 100uF are electrolytic) don't "act like" capacitors at high frequencies. For that reason, sometimes you'll see a 0.1uF ceramic capacitor in parallel with an electrolytic.
It's like what size nail or screw should you use? If you're building a bridge you have to calculate the stress on every bolt and use the right size and the right material but if are building something yourself you just choose a nut and bolt that seems "about right". The same goes with resistors and something like a pull-up or pull-down resistor can be between 1K and 10K, and even beyond that range in many cases.
A capacitor can hold a charge (a voltage) something like a battery. This is a simplification, but when I was first learning electronics I was told "capacitors resist changes in voltage" and "inductors resist changes in current." It also depends on if they are used in series or parallel... And in-series, "capacitors block DC current flow" but "pass AC".
When an Arduino input isn't connected to anything it can "float" to an unknown voltage and read high or low. The resistor "pulls-up" or "pulls-down" the input. (Pull-ups are more common in digital electronics.) When the the button/switch is activated (closed) the switch "overpowers" the resistor, a state-change is "forced" and current flows through the resistor. In this particular case, the current doesn't serve any useful purpose... It's wasted energy but that's what happens when you "overpower" the resistor (or whenever you apply a voltage across a resistor... Ohm's Law).
A pin configured as output will NOT float. So if you connect the output of one Arduino to the input of another Arduino, you don't need a pull-up or pull-down.