First of all, think of the LEDs as three completely separate LEDs that are only joined in physical characteristics, and have their grounds tied together for you.

So now you want to find out what kind of power supply you need. Every diode has a certain "voltage drop" which is pretty much constant for each diode, unlike a resistor which has its voltage drop vary with the amount of current (E=IR). You can find the voltage drop on the datasheet for the LEDs. It may be different for each different color. While you're there, find the recommended current that you can put through the LED. Let's say the voltage drop is 3V and the maximum current draw is 20mA. If you have a large, 12V power supply, that means you need to lose 9V with a resistor in line (you may want to consider something like a 5V power supply at this point, so that you don't use so much power). So, since each LED is in a series circuit with the resistor, the current is constant. That means you want to have 20mA go through your resistor as well. So now we have a voltage (9V) and a current (20mA) so we can apply ohms law to get resistance.

9V = .02A * R means that R = 450 ohms. That's the value of the resistor you want to put in series with each different LED.

The way you wire up the LEDs is simple. You take your power supply and wire three wires to it. Those three wires go to three different MOSFETs or other transistors (make sure they're rated for 12V and .02*300 = 6A). The base is connected to an arduino PWM pin with a 1k ohm resistor. The other side goes to all of the blue, red, or green legs of the LEDs in parallel, with a resistor in series with each (so 300 resistors). The other leg goes to ground. (this is assuming you have a common anode LED).

I know that's probably more than you need, but I hope that some of that will help you.