I bought a loaded pickguard and unscrewed the pickguard on there now. There are 3 wires on there that are soldered on. I do not have a soldering iron. I'm wondering if it would be a bad idea to cut and splice the wires instead or should I get a soldering iron.
Get a soldering iron.

Changing things in a guitar can become a sickness once you start, so you may aswell get one. Learn to use it on a practice piece first.
pics of gear updated on profile 11/16/09
There are 3 wires. I bought a new pickup jack prong and randomly picked one of the black wires, and also the white wire and soldered them one on each end of the pickup jack prong. The other black wire was soldered to the bracket for the tremolo bar as before.

I get a nasty buzz. What's weird is when the volume dial on the guitar is on 10 the buzz gets quieter but other than that is it is still there until a very low volume.

What could have gone wrong?

It's like the buzz changes frequency depending on whetehr or not I'm touching the strings.

I've tried grounding on pieces of bare metal I have but it doesn't help.
Lesson One: We do not "randomly pick" in electronics.
Gibson SG Special 60s Tribute
Boss TU-3, NS-2, CS-3, OD-3, CE-5
Ibanez DL10
Fender Blues Junior III
Delta Blue 5 string resonator banjo
Aria Pro II Passive P-J Bass
Aria Pro II Active Music Man Bass
Laney BC-120 Bass Combo
There were 2 black wires soldered together on the bottom of one of the tone control cases, so I took one of them and made it the ground, the other went to the input jack.

This is the only thing I could see getting wrong but if they are both soldered on top of each other how can it make a difference?

Go there and select the wiring diagram that matches your configuration.

Typically with a loaded pick guard swap, you would connect the 2 black wires (one from the jack and one from the bridge ground) to the back of the volume pot.Then the white one from the jack to one of the tabs on the volume pot.
pics of gear updated on profile 11/16/09