It doesn't get "stepped down". Let me give you a quick lesson. Imagine a standard 2-cell flashlight, where the batteries are in series. You have 3 volts across the bulb, because the two batteries' voltages add. There's 3 volts from the top battery's + terminal to the bottom battery's - terminal.
Note that there is also 1.5 volts between the middle, where the two batteries touch, and either end terminal. However, since there's no connection to this mid-point, there's only a single voltage available; 3 volts. This is what is called a "single-ended" power supply, because only one voltage is output.
Now, if you inserted a terminal between the two batteries and connected a wire to it, you could power both 1.5-volt and 3-volt loads at the same time. This is what is known as a "double-ended", or center-tapped, power supply. This is used in most modern trasistor audio amplifiers and many other circuits.
I almost all cases where a double-ended supply is used, the center point is grounded, rather than one end, so, referenced to ground, both + and - polarities of electricity is available. This is important to many integrated-circuit equipped electronics, which sometimes need both polarities for the circuit it's in.
Okay, now let's move on to the power in your home. If you can, go outside and look up at the utility transformer up on the power pole that feeds your house. (If it's an underground service, go for a drive and find one up on a pole.) On the side of the big gray can, there are 3 terminals.
Remember the flashlight? Except for the fact that this is alternating current instead of direct current, it works the same: there is 240 volts between the two end terminals, and 120 volts between the middle terminal and either end. These 3 wires come directly into your house (through the meter).
Note that the center terminal is connected to the can itself, as well as to the bare wire that runs with the other two wires to your house. This is the "neutral", which corresponds to the middle of the two batteries. The neutral is grounded, which limits the voltage-to-ground entering your house for safety reasons.
Once inside the breaker panel, the two hot wires are fed through the main breaker and on to the 2 "busses", or rails, that the breakers stab onto. Heavy appliances and other equipment use 240 volts, and regular branch circuits use 120 volts. The big question is "why?" Okay, read on:
Wattage, which loosely corresponds to work done, is the product of volts and amps. In order to do a certain amount of work, a certain amount of power is consumed. Watts' Law tells us that, when you double the voltage, half as much current is needed to produce the same wattage, or power.
If you have half the voltage available, twice as much current is required to produce the same amount of work. Because conductor is more expensive than insulation, it's more economical to supply a higher voltage and less current. Plus, the lower the current, the lower the voltage drop (loss) in the wires.
So, your large electrical loads use both hot wires (and the neutral if any part of the load uses 120 volts, like the timer and drum motors in your clothes dryer; only the heater element uses 240 volts), while your lights and small appliances use 120 volts, which means that the neutral is one of the circuit conductors.
If you look in your breaker panel, the double-handle breakers (aka double-pole) feed 240 volts (or 240/120 volts), while the single-handle breakers (aka single-pole) feed 120 volts. This is also the distinction between the 120-volt and 240-volt run you're asking about. Which to run depends on your electrical needs.
If you need only one circuit, it'll be a 120-volt run. If you need two circuits, you can run two 120-volt circuits or one 240/120-volt circuit. The difference is that (as long as the two hot wires come from the opposite hot wires) a single neutral can be shared by the two hot wires, just like your incoming electricity has.
If you need several circuits, a sub-panel run is an option to individual circuits. This means that a single cable brings 240/120 volts to a smaller panel located close to the load, which may provide several advantages: 1) local circuit control, 2) lower voltage drop over one large cable, and 3) possibly less labor.
As for 20-amp vs 15-amp circuits, this depends upon what you need to power with each circuit. Obviously, a 20-amp circuit has a third higher capacity, but more smaller circuits can be more versatile than fewer larger ones. For example, in my own HT closet, I use eight 15-amp circuits; none need be larger.
So, let's see what your system components are, and what their power requirements are. Also, tell us your forseable upgrades, like a projector, larger amplifiers, etc. , as well as present and future lighting needs. This is what is required in designing any electrical system to feed specific loads.
In addition, how far is it between your breaker panel and where the theater will be located? This is another factor when choosing between a sub-panel and individual circuit runs. Note that, beyond the sub-panel, the individual circuits will be the same: 120 volts; this is all about getting the power to the area.