I've been buying parts for new PC over the last couple of months, and the day finally arrived that I could put it together.  Here's what I got (just the parts, I'm not worried about finding the price I got for them):
 
Intel Core 2 Quad Q6600 @2.4Ghz
<a href="http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=2905631&CatId=2758> here</a>
 
KFA2 Nvidia GeForce 8800GT OC 512Mb PCI-e
<a href="http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=3441351&CatId=3670">This one doesn't look like the one in this picture</a>
 
GIGABYTE GA-P35-DS3L LGA 775 Intel P35 ATX Motherboard
<a href="http://www.newegg.com/Product/Product.aspx?Item=N82E16813128059"> here</a>
 
LITE-ON 20X DVD±R DVD Burner Black IDE Model  
<a href="http://www.newegg.com/Product/Product.aspx?Item=N82E16827106228">here</a>
 
Western Digital Caviar SE16 WD5000AAKS 500GB 7200 RPM SATA 3.0Gb/s Hard Drive  
<a href="http://www.newegg.com/Product/Product.aspx?Item=N82E16822136073">here</a>
 
NZXT ALP-001 Alpha ATX Mid-Tower Case - Clear Side
<a href="http://www.tigerdirect.com/applications/searchtools/item-Details.asp?EdpNo=3438627&sku=A406-1084&srkey=nzxt%20alpha">here</a>
 
Thermaltake Purepower W0100RU ATX 12V 2.0 500W Power Supply 115/ 230 V UL, CUL, TUV, FCC, and CB certification  
<a href="http://www.newegg.com/Product/Product.aspx?Item=N82E16817153052">here</a>
 
I believe I got all the parts put together safely.  I read about how to do it before I got all the parts out, then I read all the directions as I went.  
 
The first time I tried to turn it on, the blue lights aroung the front flashed on and off when I hit the power button, and I think I heard the fans start to spin for a half second before the lights flashed and it tried again.  
 
I unplugged it, opened it up, and messed with pretty much all the power connections to make sure I'd done them right.  Now the best I can get to happen is the first time I try to turn the system on after I plug it in, the light flash on for a split second and the CPU and video card fans want to start to spin, and then it all stops.  Now this only happens the first time I try to turn it on after I plug it into a new outlet.  If I try successive times, it doesn't do anything, but when I put it in a different socket, I can get the flash and little spin again.
 
I apologize if you guys answer this all the time and if my post is a particular pain to read.  I've been at it for a long time now, though, and I have no clue what to do, and I need to go to bed.  Thanks so much in advance for anybody who can help me.

Sounds like one of the PS rails is shorted.  There is a circuit breaker in PS that prevents damage to the board and or PS if it detects one of the power rails is shorted ie(drawing too much current)  The technical term is call "Crowbarring".
 
Here is what I would do.  Pull the motherboard out of the chassis.  Just hookup the PS and have processor, memory and Keyboard and mouse and of course the heatsink.  If the system does powerup then you know that your installation of the motherboard and other parts into the chassis is causing your problem.
 
Recommend you use some cardboard or kitchen table to do the external build of the system.

Or a bad, out of the box PSU.
 
After that, try this:
http://www.tomswiki.com/page/Troub [...] +New+Build
 
The first step is to look for anything obviously wrong. And the second step is take everything out of the case and breadboard the system. That's what pausert20's suggestion is called.
 
I do that BEFORE a build to ensure I am putting good parts into the case.

Its also possible that the BIOS is in safety mode from the factory, you will get the same result if it is, try resetting CMOS by following the instructions in the owners manual.
 
Also make sure your case PWR, and Reset connectors are correctly installed to the pinouts on the motherboard.
 
I would try this first before you breadboard the system, because if that is what the problem is then you are wasting time by pulling it all apart.
 

Did you remember to put the motherboard mounting standoffs in the case? I have seen a few people roast their systems not doing that.

hmmm.. i dont see any RAM listed there...

makes sure the 4 pin power cord is plugged into the spot (usually) located above the processor and to the left.
 
Reset CMOS, unplug unnecessary components is also a good way to troubleshoot.

1 - Check CLR CMOS jumper.
2 - Check if MOBO is not touching the case directly.
3 - Check 115~230 switch on the PSU (gotta love that one)
4 - Check power contact on ATX power connectors (20+4pin AND 4pin)
 
(try the above with only CPU+RAM+VGA connected to the mobo. You don't even need the HDD. On the Outside, Keyboard, Monitor and POWER, the latter is very important )
 
If all above is correct, test your PSU. If you can't get ahold of another machine to try it on, short the green pin with any black pin (on the PSU ATX 20-pin), while the psu is ON. That should start the PSU and it should keep running. If you don't trust me on this one, see
http://en.wikipedia.org/wiki/ATX
 
Anyway, definitely a power issue, on my opinion. And randomly switching outlets won't solve that. If none works, try a different PSU.
 
Oh, and don't forget to connect the internal PC speaker, cause it will make siren noise if the problem is either the CPU or the RAM. That would help sorting things out.

I haven't tried anything yet, but I just wanted to thank you guys and address harsher' RAM comment....I did actually buy RAM, but I was too tired to actually remember to post.  Also, sorry about botching hyperlinks in my original post.  That, too, slipped under the radar of a tired, frustrated man.
 
The RAM is  CORSAIR Dominator 2GB (2 x 1GB) 240-Pin DDR2 SDRAM DDR2 1066 (PC2 8500) Dual Channel Kit Desktop Memory
http://www.newegg.com/Product/Prod [...] 6820145043
 
Thanks again, I'm going to go try those things, and if I'm lucky, my next post will be from my new build!

if it still doesn't work... start unconnected components...
hard drives, DVDR,  
with no video card should be error beeps...
no ram should give you error beeps as well...
just bare board with just power but no ram , CPU, or video card should give you error of no CPU installed...
if you don't get those beeps it could be bad board or bad Power supply...

Ouch.  That DDR2-1066 may not work reliably on that P35 mobo.
 
But yes, the main problem sounds like a rail is being shorted.  The mobo needs to be on standoffs, and ONLY the standoffs underneath screw-holes should be installed.  Make sure an extra screw isn't trapped under the mobo, shorting it out.

Poster still didnt mention stand offs.  Did you use standoffs?  Did you use the correct screws to mount the mobo?  
Theres a million reasons it could not be working, now your going to learn what building/fixing computers is all about......troubleshooting.  
Are you sure you have the 20pin connected to the mobo?  Do you have the 4 pin cpu power cable connected to the mobo by the cpu socket?

To address the previous concerns:
 
A) If you check the specs on the motherboard, it's rated for 1066
 
B) Sorry I forgot to address it earlier, but yes, I used standoffs
 
and now I'm actually making this post from the new computer!!
 
I figured out that I was quite the silly boy and thought that there needed to be a power connector in the secondary system fan jumper on the motherboard.  I realize now that that is for a fan speed controller.  As you can tell, I'm a first-time builder, and thank goodness for safety features on components for idiots like me.  I unplugged everything besides the power to the mobo, CPU, and VGA, and the fans stayed on, so I played with connecting from my molar cables one by one until I isolated the problem.
 
Thanks again for everybody's help in getting me started on my brand new pretty computer!

My GA-P35-DS3L and Abit IP35 PRO should be good for 1066 too, but aren't.  Oh, in each case it seemed to work for a while, then the corruption started.   I've read in other forums about compatibility issues.
 
The fan jumper is not for a speed controller, it is for a fan or a speed SENSOR wire (typically from the PSU to monitor its fan).