PA System Will Not Work

Change one thing at a time until you find the problem, but given your comment about the mixer, I'd start at the very beginning of the signal chain. No flashing lights on the mixing board = no signal coming through from the mic. So, problem's likely either at the mic, mic cable, or somewhere on (or in) the mixer. If it's a condenser mic, make sure you're providing it with 48v phantom power from the mixer. Check your input gain settings at the mixer too. Try other inputs; try other sources besides the mic.

It looks like everything was connected correctly (although I couldn't see where the cables from the mixer to the amp were plugged in on the mixer. They should be to the "main out" positions.). You are good at the speaker. The other connectors that you pointed to are Speakon connectors, You can use them or the 1/4 inch plugs. Either is OK.

Now, the mixer.
I'm with Slej, that's the first place to look. Or the mic or cable.

First, for the protection of your ears shut the amp off for the next few experiments.

For anyone playing along at home, here is the mixer manual http://www.behringer.com/EN/downloads/pdf/1622FX_P0557_M_EN.pdf
and here's a big picture of it:

Now, your mic is an SM58 right? So it's a dynamic mic. That means you don't have to think about phantom power.
What is the "gain" control set at (it's the one closest to the XLR plug that your mic is connected to)? Each input has one, and it's used to get the level in the ballpark. Start talking into your mic, and turn the gain control up (clockwise) until
(A) you notice the "clip" light start to flash with your voice.
(B) you run out of adjustment on the gain knob.

If B, then there is no signal getting to the mixer (or it is faulty. Try on a few other input channels)
Suspect the mic or cable. Try another if you have them.

If A, then signal is getting into the mixer. Turn the gain control down a bit until the clip light *just* stops flashing. This is a good place to leave it.

Now that we know the signal is getting in, lets see where it goes.
Each channel strip has a "mute" button. It should be up, and the mute light should be off. (obvious sounding, I know, but it catches some people)
There are a set of routing switches beside the sliding fader at the bottom of the channel strip.
For a basic setup, the "main" should be pressed. That will allow the signal to get from this channel to the main mix output fader on the right side of the mixer.

Now, push the channel fader up to about the middle of it's range.

At this point, is you start moving the main mix fader up (and keep talking to your mic), you should start seeing some action on the output level lights on the right side, just above the main mix fader.

If you do, then pull the main mix back down to zero, and turn the amp back on.
Now gently slide the main mix up and listen...

Sound?
Yes: good
No: is the level control on the amp turned up? Are the cables connected on the amp correctly?

Kent, I think that was possibly the most helpful answer I've ever been given. Thank you so much.

I got the PA system working but now I've got a new problem. It's really more of an annoyance than anything but I've noticed the sound coming through the speakers is crackling a little. I'm guessing this may be from the $15 XLR cables I bought for the main out to the amp; either that or the GLI speakon cables I bought in a pair at $20 (MSRP was apparently $70). Any thoughts on this? I still don't have all the amplification settings down just yet but I'm guessing the cables are the biggest problem.

Bad cables and connectors are most often identified by wiggling them, one at a time.

If they are all new cables, though I'd be surprised at noise. A new cable usually works or 100% fails.

If the crackling is still there with the mic muted, then it's not the mic cable.

Is the noise in both speakers, or just one?
If you swap the mixer->amp cables (move the left to the right, and right to the left) does the crackling move to the other speaker, or stay where it is?
Same with the speaker cables.
(power off the amp when you unplug a speaker cable - depending on the amp, it may or may not like pulling the cable powered on. And it's just good practice.)

By the way, what was the original problem? and what solved it?

Turned out my system was fine. I either bought a low quality 1/4-1/8 for my iPod or it just wasn't plugged in well enough. Works fine now. My original problem was my system didn't want to give me any sound through either my mic or my iPod. Turned out it was just the "Main" button was off and the "Trim" knob was down too far. I also turned on the "Clip" button and that seemed to help but I think it would have worked without it.

Now I'm just trying to figure out why my computer won't play back my recordings. My computer is registering a signal but I get no sound on playback (and I can watch a video on, say, YouTube and it's fine). I've got it running RCA out to a USB interface.

The button adjacent to the clip light is the solo button.
It will send just that channel to the headphones (without affecting the main mix).
This lets you "zero in" on one mic during a full mix, to hear what's going on. Useful for making channel equalization tweaks, or hunting for odd sounds (which mic is picking up that wind noise?)

Well man, I can't even thank you enough for all your help. I've got everything up and running and have started some test recordings.

... now just to find a good sound guy... ha ha.