How do I figure out the exact voltage and current needed for a part?
If you have only the power rating, you are stuck.

If you have the resistance in ohms as well, you have enough information via either of the equations:

$$P = \frac{V^2}{R}$$

or  

$$P = I^2  R$$

So given a 2 Watt 8 ohm speaker, 

$$V^2  = P \cdot R = 16$$

so V=4 volts, and I = V/R = 0.5 amps

Now for a speaker you also need to know if that was the peak power rating or the RMS (roughly speaking, average) power. If it is 2W "rms" that means 4 volts rms, or 2.8*4 = 11.2 volts peak-peak, which suggests an amplifier running off 12V DC.