1. I don't know how much their external power supply should be (Just 6V? How many Watts?)
Power supplies are usually rated with volts and amps. Check the output-current rating for the power suppy.
But FYI, power is calculated as Voltage x Amps.
300mA x 6V is 1800mW or 1.8 Watts.
Watch out! Sometimes the "wattage rating" applies to the power pulled from the wall, which is usually more than it puts-out (since power supplies are less than 100% efficient and the wall-power rating is for UL or other safety rating, and it might be over-stated).
With a little algebra, the power formula and Ohm's law can be combined to get Power = Voltage squared/resistance or Current squared x Resistance.
2. I am TRYING to use ohm's law to figure out which type of resistors I'll need... but something doesn't seem right... I suck at math
The resistor in series with the transistor base, I assume?
Transistors typically have a current gain of around 100. But for a switching application, let's assume the gain is only 20 to make darn-sure the transistor switches all the way on. We can also assume/approximate that the whole 5V is dropped across the resistor (although it's more like 4.3, with 0.7V dropped across the transistor's base-emitter junction).
300mA/20 = 15mA base current.
5V/15mA = 333 Ohms Since we made the gain assumption of 20, anything in that ballpark will work.
You don't want to make the resistance too low, or you'll have too much current (from the Arduino, throught the resistor, and into the transistor base, not through the motor), and things could overheat & burn-up.
If you make the resistance too high, the current into the transistor base will be too low, making the current through the collector, emmiter, and motor too low, making the motor run slow, and possibly overheating the transistor by switching it half-on, where it will have voltage across it, and current through it at the same time (dissipating power).
3. Capacitors/transistors. I am in the dark here. Will PN2222A transistors work?
I think so... I'm not sure. Check the current rating for the 2N2222. Don't worry about the transistor's voltage rating. Any
transistor will be OK with 12V.
25v 220uF Capacitor?
I don't know... Do you need a capacitor? Where does it go?