Well, parallel wiring would be my choice, so that on the off chance that you have a bad LED or one fails, you'll be able to tell, and still have a working circuit.

Yes you would need different resistor values for each scenario. Make sure you know the typical voltage drop across each of your diodes, their max. power dissipation, and their reverse-bias breakdown limits, then just uses Ohm's and Kirchoff's laws together to get the right currents and voltages.
^ Wow, talking about losing me in that second paragraph!

So - voltage drop? Is this something specific to each kind of diode? The name suggests to me how many volts it saps from the current?

What am I trying to achieve by using the two laws together (I only know Ohm's law, not Kirchoff's. I will look it up)? Sorry, I'm a newb to all this electronic stuff.
Ok, all diodes drop some voltage across them, but unlike most components, they won't just dissipate as much power as is available. Instead, they won't drop more than a certain amount, and below that voltage, they have a very non-linear response. For simple silicon diodes, it tends to be just under 0.6V, for germanium, a little higher, ~0.6-0.7V. Each different type of LED will drop a different voltage, so make sure you know what type you've got and read up the specs.

All components can only dissipate so much power before breaking (usually by melting or burning out), so you need to make sure the total resistance is such that it keeps the current power dissipated in the LED's (P = VI) below what they can take. So now you know both the voltage across the diodes and the current through them, so you can work out the total resistance the circuit has to have using Ohm's law.

Kirchoff's laws are for calculating resistances in series and parallel. they're fairly simple.
For resistances in series, total resistance R = R1 + R2 + R3 + ... etc
For parallel, 1/R = 1/R1 + 1/R2 + 1/R3 + ... etc
You need these laws to make sure that you get the right resistances in the right place. It's fairly simple to combine the two for series/parallel wirings as well.
I want to run two LED's in a parallel circuit. Here are their specs. Could someone help me to work out the resistors I would need to use, please?

LED 1:

Forward voltage: 2.5V
Forward current max.: 25mA
Reverse voltage max.: 5V
Wavelength @ peak : 565nm
Power dissipation PT: 105mW
Light output min.@ 10mA: 5mcd
Light output typ.@ 10mA: 20mcd

LED 2:

Forward voltage at If=20mA: 4.0V
Forward current max.: 30mA
Reverse voltage max.: 5V
Wavelength @ peak : 430nm
Power dissipation PT: 105W
Light output min.@ 20mA: 50mcd
Light output typ.@ 20mA: 150mcd
Ok, your green LED (LED2) is rated to 30mA, but they give you the forward voltage @ 20mA, so it's probably best to aim for that sort of region.

What's the output of your power supply?