Step 1: The Parts
1 LM317T Adjustable Voltage Regulator - 276-1778
This is the adjustable voltage regulator. It takes input from two resistors (R1 and R2) and then ratchets the voltage down accordingly. I recommend you take a look at the datasheet if you want to learn more about this part.
1 0-5K Linear Potentiometer - 271-1714
This is R2, and will allow us to control the voltage output.
1 560 Ohm Resistor - 271-1116
This is R1.
2 1N4001 Diodes - 276-1101
There are two diodes to protect against short circuits. D1 will protect the regulator from the capacitors discharging if the input power is short circuited. D2 will protect the regulator from the capacitors discharging if the output power is short circuited.
1 .1 uf Capacitor - 272-135
This capacitor (C1) acts as a smoothing capacitor. It should be only a ceramic disk capacitor.
1 10 uf Capacitor - 272-1013'
This capacitor (C2) improves the transient response of the regulator. It should be electrolytic.
1 PCB mount SPST switch - 275-645
Allows you to turn the power on and off without unplugging the wall-wart.
1 PCB mount terminal strip - 276-1388
This is mounted directly to the PC board and is an easy way to connect your power supply to many different circuits and components.
1 12v Wall-Wart
Provides the power to the circuit. RadioShack has a nice selection, but I recommend salvaging your own as I did. Anything will work as long as the output current is no more than an amp. I choose one that has an 800mA output, but anything over 500mA should cover most basic electronics projects.
1 Small Perfboard - 276-148
This particular board is the perfect size for this circuit, and my layout is based on it. This is a perfboard, but if you wanted to make your own PCB, feel free to use the attached EagleCAD schematic to generate your own layout.
1 Heat Sink - 276-1368
A good precaution. The regulator has built in protection to prevent it from burning itself up, but it does that by limiting current. If you didn't have a heat sink, you might find that you have less current output than you expected. Any piece of metal will work as long as you can attach it metal-to-metal on the tab. Even a large alligator clip will provide decent heat dissipation.
Step 2: The Tools
Not a soldering gun. Soldering guns make semiconductors cry tears of melted plastic.
Hot glue doesn't count. I've seen it. No joke.
Not required if you don't think you're going to make mistakes.
Small flat-headed screwdriver
To tighten the screws on the terminal strip.
These are those funky things that hold stuff while you're soldering. These are useful for way more than electronics.
Solid core wire
You need this to create the traces. It must be solid core!
Step 3: Breadboard
Step 4: Dry Fit the Components
At this point, your also going to want to bend the copper traces. Use the wire strippers to strip all of the insulation off of a length of wire, and bend them to the correct length. Bend all copper traces except for the ones that lead to the terminal strip. These are more easily added after all the other components have been soldered.
Step 5: Solder Stuff
Step 6: Solder More Stuff
Step 7: Quality Control
Step 8: Moment of Truth
Step 9: Possible Improvements and Modifications
VOUT = 1.25 * ( 1 + ( R2/R1 ) )
The value of R1 should be between 10 ohms and 1000 ohms. Anything higher and the voltage regulator won't behave. If you decide to make any changes to the circuit, you should refer to the datasheet for the finer details. This site is another good reference for using the LM317T.
Ideas from the comments
-You could enclose the entire circuit in a plastic project box. That would prevent the back side of the board from shorting out if it came in contact with a metal tool.
-You could buy a multimeter that is to be used only for tuning the output. Cut off the probes and solder the wires directly to the outputs for a permanent solution. If the multimeter had interchangeable probes, you could buy an extra set for use with other projects.