FALKEN said:
Tonight, I will try and build the power supply, and see about adding some TL082's to this thing. If those work out, I will order up something nicer. Are there any safety precautions I need to take so that I don't kill myself?
The only thing really dangerous in a low-voltage power supply is the primary of the transformer when it's plugged in. Everything on the secondary side is low voltage. However, if you screw up a rating of a component, you could pop a cap, or get a resistor or regulator really really hot. Keep an ABC fire extinguisher handy

and also be careful about what parts you touch while you are messing around. Burning your fingers won't kill you, but it does hurt!
What should I get at radio shack to build the supply on? I'm assuming you can't run wall current through one of those pcb's, right?
You can, but you don't need to. The line from the wall should go into a fuse, a switch, then straight to the primary of the transformer, usually all done point-to-point, but it can be on a PCB.
Also, do you use the same ground for the power as the signal ground?
The power ground wire is screwed to the chassis; signal ground often is attached to the same point, although unbalanced gear often has signal ground isolated from the chassis. I'm not an expert on grounding theory; it gets complicated sometimes
Finally, what kind of impedance does the op-amp need to see in order to function properly? should it be the same for the inverting input and the non-inverting input?
The input side of a FET opamp is very high impedance, so it's usually up to you where you set impedance with a resistor shunt to ground. 10K is typical for line input.
Edit: Just had a look at the TL082 datasheet, it's limited to 20mA output, so: A FET opamp output can go down to around 1000 ohms, and often you will find resistors in series with the output to protect against shorts to ground.
Also, is it a good idea to build the PS in a separate box, so I can use it for other projects?
Thanks!
Sure, you could even use an XLR connector to make switching easy, pin 1 ground, pin 2 +15V, pin 3 -15V.