Well, I got it running...

[missizzy]

After connecting all the cables this morning and getting the power source more or less screwed in place, I set to work trying to connect the peripherals, found there was no cable or cord for the monitor it was earlier decided I'd use, connected the monitor from this computer instead, took a deep breath, and pressed the button. It didn't turn on. I took the panel back off, plugged the CPU cable back in, tried again, looked again, plugged in several tiny things that had also gotten unplugged, tried a third time, and what do you know, it actually worked. Installing Windows on it was its own headache, but that was eventually accomplished as well. I have successfully built a new, working computer.
Except the sound's not working. I didn't have speakers, but when I tried to plug my headphones in instead, the computer wouldn't read them. I even opened the thing back up again to see if I'd maybe plugged the HD audio cable into the wrong place, but I really don't think I did.
I have now ordered a new cable and plug for the monitor, and new speakers. I could try connecting the speakers from this computer, but my will for it is spent for today. Once they arrive, I'll see if they produce sound. If they don't, I can call my normal IT people. This, at least, is a problem they can and will help me on.