[Jess_hp]

I am trying to replace a standard electrical outlet in my bathroom with a GFCI outlet. There are also two light switches, one of which appears to require power through the outlet, although there are 3 sets of wires running into the electrical box. When I tried wiring the light switch to its own set of wires (not through the outlet) it did not work. I am able to get the switch to work when wired to the new GFCI outlet, but the outlet itself will not reset or turn on at all. I have tried every possible wiring combination. Any suggestions?

[twangwe]

I would think with a voltage tester you can determine which of the 3 lines is the feeder(supply) line. This line goes to the line side(which should be clearly marked) on the GFCI. Your light switch will go to the load side(also protecting it). Unsure of what your third line is unless it is a receptacle branch. Which would also go to the load side. How was the old receptacle wired?

[hkstroud]

First, this is not going to be easy to figure out. If you had an outlet that was not a GFI, the wiring is probably fairly old. Prior to GFIs you could not have an outlet within 6 foot of water source. The old outlet was probably done by a prior owner and who knows how things were done. The outlet may have been wired through the switch and was only live when the switch was in the "ON" position.
You said,

there are 3 sets of wires running into the electrical box

But I don't know which box you are talking about.

Describe the wiring in each box Describe like this.
A wire is a single wire. A cable is two or more wires in an outer covering. There are 2-wire cables (black and white) and 3-wire cables (black, white and red). We don't count the bare ground because it is not a conductor and is not part of the circuitry.

Describe the number and kind of cables in each box.
I assume the two switches are in the same box. What do they control?

Do you have a meter to test things with?
Do you remember how things were wired before you started?

[Missouri Bound]

but the outlet itself will not reset or turn on at all

It's wired incorrectly. Keep in mind that a GFCI will not test or reset unless there is power to it.

[inla_bomber]

It's easy to fry a GFCI. That may be why it will not test. If the wires in the box are causing you confusion, try a regular receptacle outlet first, until you figure the dynamics. At least then, you won't be burning through your money.