Ok... After considering the good information that everyone was so kind to provide me, I have decided to try something else.

How would I modify a circuit similar to the one below to produce 12-24v rather than 0-24v? It doesn't have to be exactly 12-24v. It is just a ball park figure.
This is beyond my current knowledge of circuits.

 

What kind of amperage are you looking for?

 

Whatever I can get out of the chip. It is rated for 1.5A so anywhere from 1A-1.5A is good.

 

Set R2 to produce 12 volts. Measure its resistance and find a fixed resistor of that value. Place that resistor between R2 and ground. Now the regulator will vary between 12 volts and the maximum.

 

R2 = R1 * ( (vout/1.25) -1 )
R2 = 240 * ( (12/1.25) -1)
R2 = 240 * 8.6
R2 = 2064Ω

 

Thank you retched, beenthere, and tracecom! I appreciate the help. I think that I have asked a question in which I should have already known the answer.