Your question is very general, and so is this answer.
When a power plant creates power like the Hoover Dam, it can provide 2.07 GW of electrical power. My question is what does this mean? I assume from Faraday’s law that the induced voltage across the generator coil produces an current, and this combination (P = VI) is the actual power but I'm sure that my thinking is naïve. Can someone roughly sketch out how the electrical power of a power plant is computed?
From a mechanical perspective
"2.07 GW" means that the peak output of the power plant is 2.07 GW. This is most likely a series of smaller units, say 20 × 100 MW units = 2.0 GW.
The generator is a converter of mechanical energy into electrical energy. So to generate 2.07 GW of electrical energy, an equivalent amount of mechanical energy has to be provided. In the case of the Hoover Dam, the mechanical energy is provided by water falling to a lower elevation, giving up its gravitational potential energy in the process.
From this perspective, you can think of the maximum electrical output power of a power plant as the maximum rate at which it can convert mechanical energy into electrical energy, factoring in the efficiency of the conversion process.
The rate at which mechanical power is generated is a mechanical engineer's problem. For a hydroelectric station, the mechanical power would depend on the water pressure, turbine size, and various design parameters. For a wind turbine, the mechanical power would be set by the radius of the blades. And so on.
From an electrical perspective
Yes, the electrical energy produced follows Faraday's Law and Ohm's Law, though for an AC system, V and I are sinusoids which may not be in phase, and P ≠ VI. Rather, apparent power (volt-amperes) S = VI, and real power (watts) P = VI cos ɸ.
Other complications include electrical losses (per Ohm's law) and magnetic losses (eddy currents induced in metallic parts).
If possible, what kind of voltages and currents are power plants producing before the Step Up transformers? Of course, this varies from one power plant to another.
Regarding typical voltages
In my experience, small generators (i.e. diesel gen-sets) generate directly at the utilisation voltage, say 415 V here in Australia.
Larger power station units generate at a medium voltage like 11kV before stepping up to transmission voltage, i.e. 132 kV.
I imagine a medium voltage like 11kV is preferred vs. a high voltage like 33kV, because less insulation is required on the windings and the rotating parts may be physically lighter.
Regarding typical currents
An aeroderivative gas turbine, i.e. the General Electric LM6000, is typically rated about 45 MW and might have a 60 MVA alternator attached to it. Calculation of the three-phase line current at 11kV is left as an exercise to the reader. Don't forget your √3.
A coal power station unit might be rated 400 MVA at 22kV. See "Tarong Power Station" in QLD, Australia, which consists of four large units like this. Again, calculation of the line current is left as an exercise.
Note: I am at home and hence don't have access to my reference material at work. The above numbers are indicative, so treat them with a grain of salt.
If you are curious as to the exact operating principles and theory of an AC generator, I would encourage you to look up a textbook on electric machinery. My personal favourite is Mulukutla Sarma's Electric Machines. Check your university library for a copy.
Best Answer
Yes, exactly right.
To simplify, imagine a single 2-wire AC circuit (not complicated 3-phase.)
First, the utility company dynamo will charge two wires opposite, like a very long capacitor. One wire is temporarily positive, the other negative. Next, your motor inside your washing machine will "discharge" this long capacitor, taking some energy from the e-field which exists in the space between the long wires. (The dynamo increases the voltage between the wires, and simultaneously, the motor slightly the voltage.) The dynamo "charges" the capacitor with energy, while the motor "discharges" it.
But also, the utility company's dynamo will create a current in the entire circuit, including the miles-long wires and also inside the AC motor coil. The entire circuit stores energy as a magnetic field (it's a 1-turn inductor.) The dynamo increases the current in the circuit, while the motor simultaneously decreases it slightly. The dynamo "charges" the inductor with magnetic energy, while the motor "discharges" it.
Together, the above two effects are allowing electro-plus-magnetic energy to race along the circuit, flowing from dynamo to motor. The circuit behaves as an inductor-capacitor ...also called a "transmission line."
In other words, the dynamo injects EM energy into the entire system. Then the distant motor withdraws energy from the entire system. The EM field-energy then flows across the system at the speed of light. (But electrons themselves travel slowly.) Electrical energy is wave-energy, while the electrons inside the wires are the "medium" which guides the waves. (Don't forget that sound-waves travel fast across great distances, while the air-molecules themselves just vibrate slightly. Waves versus medium.)
Here's the secret to understanding energy in circuits: all circuits are EM waveguides. They are radio transmission lines, even when the transmission frequency is 60Hz, and even when the frequency is DC or 0Hz. In other words, transmission lines have no lower limit to frequency. The same mathematics which applies at 1MHz also applies at 1Hz and at DC. In engineering classes, many of us miss the fact that our fields/waves textbook wasn't describing RF transmission lines ...it was describing all transmission lines, including 60Hz power grid, and including the conductors inside a DC flashlight.
Note that with car engines and drive shafts, the energy travels as sound-energy of very low frequency. The metal atoms in the drive-shaft are the "medium" for this mechanical/acoustic energy-flow. The metal atoms don't flow along the drive-shaft! But the kinetic energy does flow along the drive shaft, going from motor to wheels (and ...it travels at the speed of sound in steel!)