I shall do the best I can to explain things for ya:
Quote: "What does a resistor actually do? Most tutorials say it "resists" current flow, but what does that mean?"
I have a great analogy for you; many electronics schools use this analogy to teach. Basically, you can think of any circuit as a series of pipes that water can flow through, often in circles (by circles, I mean the water keeps cycling through the circuit over and over). Voltage can be thought of like water pressure that is trying to move water through a pipe. Voltage is essentially the force that tries to move electrons, and the greater the voltage, the greater the force trying to move the electrons. Any time electrons move through a substance, we refer to that as electricity. An electrical current is the movement of electrons through a conductor, often a wire or other electrical component.
Now, as water flows through a pipe due to pressure (just like our voltage), it experiences resistance. In fact, resistance is the reason that water doesn't flow through the pipe infinitely fast. Electrical current can be thought of like the current of water flowing through a pipe. The more electrons flowing through a component or wire, the greater the electrical current.
All electrical components have some resistance; that's an unfortunate fact of practical electrical engineering. So to answer your question, resistors are basically components designed specifically to have a certain amount of resistance, often way more than a wire would. In our analogy, a resistor could be thought of like a constriction in a pipe; water will have a harder time flowing through that constriction than through an open pipe. The resistance of a resistor is measured, as you probably know, in Ohms. The greater the Ohm count, the more strongly the resistor will resist the flow of electricity. A resistor of a greater Ohm count could be thought of like a smaller constriction in a pipe.
Quote: "If I were to stick a voltmeter on the other side of it, would the voltage still be the same?"
This is a rather vague question, for reasons that would be sort of hard to explain.

It depends where exactly you measure the voltage from the resistor. You could measure across the resistor, between the resistor and another component, etc. I won't get into that for now.
Quote: "Does a resistor "set" the current to a specific value regardless of available current?"
No.

So back to our analogy, if you look at a water pipe, there will always be two factors determining how much water is flowing through it at any time. First, water pressure. The greater the water pressure, the greater the water current, and vice versa. Second, the pipe's resistance to the flow of water. Opposite of pressure, the greater the pipe's resistance to water flow, the smaller the water current, and again vice versa.
If we flip that analogy over to the electrical side, we can infer that there are two factors that always determine the amount of current flowing through something at any given time -- voltage and the resistance of that component. The greater the voltage, the greater the current, and vice versa. The greater the resistance, the less the current, and vice versa. Here is a handy and super simple formula that allows you to calculate the amount of current flowing through any component, given the voltage across that component and the resistance of that component.
I = V / R
I is current, in amperes. V is voltage. R is resistance, in Ohms. Simple as that! Voltage divided by resistance gets you your current.
Quote: "So say I have a small (watch-battery-sized) 5V battery connected in a circuit to itself. In that circuit I stick a 1K [insert Ohm symbol here] resistor... that makes it 5mA of current, right?"
Yes! You must know that formula I gave you. You simply took the voltage across the resistor (5 volts in this case, which even with a 5 volt battery
may not always be the case, but that is more advanced so I will spare you that confusion for now), and divided that by 1000 Ohms, which left you with 0.005A, or 5mA.
Quote: "So what happens if I replace the watch battery with a much larger, say, ATV battery (still 5V). If I understand correctly, it would be capable of providing a far larger current... so would the resistor still bring it down to 5mA?"
Yes! Indeed, the battery would be capable of providing more current, but the resistance and voltage are still the same, so the current will remain just the same. I haven't mentioned this yet: in our analogy, you can think of a battery as being like a water pump. A battery of a greater voltage is just like a pump that imposes a greater pressure on the water it is trying to move.
Quote: "Does a resistor only change the current after the point in the circuit where it's placed?"
No. Current is always the exact same all throughout a closed circuit. There are some exceptions to this, but those are more complex circuits where current can branch off and take different paths through the circuit. We won't get into those for now.

Just like our water analogy, if you have a pump and a pipe that connects its input to its output, and somewhere along that pipe you place a restriction (just like a resistor), the current will be the same all along that pipe.
Quote: "To be clear - when you lick a standard 9V battery, it's pushing all the current it can provide through your tongue, right?"
Not necessarily. The human tongue has a fair bit of resistance, so even with 9 volts, the current across your tongue will be very small when you lick a battery. It feels pretty intense because your nerves are very sensitive to small current.

I am not quite certain what that current would be, since I don't know the resistance of a tongue (and I am not sure I want to test mine

), but I would suppose it's only a few mA. A 9V battery is capable of providing far more current than that, and will indeed provide more current if connected to a circuit of lower resistance.
Quote: "So if you licked a car battery (using jumper leads or something) you'd get a far larger, probably dangerous shock?"
Also not necessarily, but I wouldn't recommend it. A car battery is much larger, and it can deliver WAY more current than a 9V battery can, but that doesn't mean it always will. Its voltage is not that much higher; only 12 volts! It would be a more significant shock, yes, but probably still not dangerous. In fact, if we do the math, we can calculate that the current across your tongue with a car battery will only be 4/3 the current across your tongue with a 9V battery.
I hope I have explained things well.

If you want an even better explanation of the analogy I gave you, check out the site I have linked below. That article also describes the analogy for various other electrical components, so when it comes time for you to learn about those, hopefully this article can be helpful then as well.
http://en.wikipedia.org/wiki/Hydraulic_analogy