In that case, we'll assume it's unregulated (most are unregulated). The output is only approximiately 9 volts. If the current draw is lower than the 500 mA specified, or the AC line voltage is higher than 120V, the output will be more than 9 volts. Conversely, the output will be lower if you load the transformer with a greater than 500 mA, or the Ac input is lower.
LED's operate with a approximately constant voltage drop, as you see from the specification - nominally 3.3 V for the blues and whites (whites are just blue LED's with an extra coating), and nominally 2.0 V for the reds. The brightness and the operating point for the LED's is determined by how much current is pumped through them; again from the specs, 20 mA is a good figure. Lower than that and the LED's will be dimmer than they would be otherwise; much over that and they'll run hot and their life will be shortened considerably.
The way to power an LED is with a constant-current supply, set at the desired operating point. A true constant-current supply is spendy, so most of the time a quick 'n dirty alternative is used instead. Namely, a resister is used between the LED and a conventional constant-voltage, or near-constant-voltage, supply. Since the votage across the LED is approximately constant, and the output from even an unregulated power supply is approximately constant, the voltage drop across the resistor will be constant, and thus the current will be constant. The "gotcha" is to ensure that the voltage of the power supply is high enough above the voltage across the LED, so that the voltage across the resister stays approximately the same, and thus so does the current.
In this case, you want to limit the number of LED's in series to keep them below 6 or 7 volts or so. For example, 3 reds would be nominally 6.0 volts, and 2 blues or whites would be 6.6 volts. For a 20 mA current draw, the resistor should be (9-6)/0.02 = 150 ohms for each string of 3 reds, or (9-6.6)/.02 = 120 ohms for each string of 2 blues/whites.