12 and 5 volt leds are just normal leds with a resistor installed on one of the leds or wires. They may be more bulky.
I think your ohms law is a bit off.
All diodes including those that emit light have a voltage drop. This drop in voltage has to be taken into consideration when running leds.
Lets use ledcalcs numbers
Red and green: 2 volts
Blue and white: 3.0 - 3.5 volts
Now some green leds will be 3.0+ volts. This should be on the specs.
If you are just getting into this getting a breadboard is a good idea since you can test things before installing them.
Moving on to resistor values.
Voltage - Forward = voltage the resistor needs to drop at a certain current.
So lets say you have 12 volts and want to run 2 x 3.2 volt leds
12 - 3.2 - 3.2 = 5.6
Now 5.6 / current in amps(I will use 17 milliamps)
5.6/0.017 = 329.4117647058824 with the nearest at 330ohms
LEDcalc.com does all this for you(no math needed, but math never hurts you).
Forward voltage is important for getting the right resistor value. It also changes with current for this reason, I tend to over-estimate a bit and run lower current. If you want to MAX it out you will need to test with the leds you want to use.
Here are the forward voltages of some leds i had laying around here(yeah yeah my meter is connected backwards, but it still works). Do not worry about other stuff on the breadboard because it is a current source so no matter the forward voltage the set current will always flow.
@ 12 mA
@ 18mA You will notice the white now has a higher forward voltage. The others moved up a bit as well
Please remember you do NOT have to connect the leds to the fans own cable, you can use an old Y cable/molex directly from the power supply instead.