Is my CPU going to "bottleneck" my GPU?

The match between video card and your system is more directly driven by your monitors resolution and refresh rate. The CPU will generally determine the max FPS you can have at any given moment in a game, and the higher the resolution of your monitor the more GPU power is needed to maintain any given FPS target.

What is your resolution?

1440x900.

A 570 will probably have more power than your system can use in many games, but in the most demanding games, it will be needed power. What games are you playing?

Skyrim, BF3, FO3, FONV, MW3, ME3. Games like those.  

It will be worlds better than what you have now. GO for it! If it bottlenecks, it wont be by very much. YOu should be more worried about whether your power supply is up to the task.
Of the games you mentioned, all will run fine at that res with a 570.

A GTX 570 is VASTLY overkill at such a low resolution resolution. Even a $150 Radeon HD 6870 is more than you truly need, and an HD 6850 is more appropriate.

As was said, your power supply is the major concern. An older pre-built system with integrated graphics probably has some 300 W no-name piece of garbage with no PCIe cable. Depending on what video card you end up with, you're also probably going to need to invest in a good quality (Corsair, Seasonic, Antec or XFX) PSU at about 500 W. Stay away from cheap brands like DiabloTek, Coolmax, Raidmax, etc, unless you also want to invest in a fire extinguisher.

The good news is that an HD 6850 + Corsair CX500 works out to a little over $200. A little less, after rebates.
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Also, make sure your case has enough clearance to allow a 8.66" PCIe card. It probably does, but pre-built systems can be a total crapshoot.

Your cpu will bottleneck gtx 570,trust me my friend,good match for you procesor would probably be radeon 6850 at best,or nvidia gtx 550ti

If I did buy a 6850, what kind of FPS would I be expecting on some of the games I listed and at what detail? Thanks. Sorry for the trouble....

Like said by bystander, some games will be overpowered by the GTX 570 at such a resolution and some other will just be OKAY with the GTX 570 providing you playable FPS such as Metro 2033, Crysis 2 @ ultra quality.

In CPU intensive games and online multiplayer games with huge maps, you'll be bottlenecked by your CPU, most RPG games such as Skyrim & WOW depend more on CPU power so you'll have to upgrade both your monitor and your CPU in the near future, there's no problem going with the GTX 570 right now.

So what should I choose if I probably wont upgrade my CPU/monitor preferably anytime soon as money is short? The 6850, 6870, or GTX 570? Also, right now I can only play Skyrim on the very minimum settings, and lowest res windowed mode at 8-20 FPS. Is this normal?

For an integrated GPU, yes that is normal. If it were me, I'd go for the 570, because I like good FPS and high settings in the most demanding games, which in your case, will be BF3. It would also be one less thing to upgrade in the future.

However, i believe most would probably go with the 6870. It'll give you pretty decent settings in BF3, Metro 2033, Crysis 2, The Witcher 2, and most games will be better balanced.

If I bought a 570, what PSU should I get? Also considering future upgrades, because if I upgrade my CPU later I may need more watts? Right? And thank you everyone so much for the help so far.  

EDIT: Also, does my mobo matter?

That's normal. Your current video hardware is integrated, low-end video hardware from four generations ago.
I stand by what I said: go with an HD 6850 and good 500 W PSU as long as you intend to remain at 1440x900.

Of the games you listed, the only one where an HD 6850 may be pushed at 1440x900 is Battlefield 3. My personal experience is that I can run BF3 at 45-60 FPS on a GTX 460 (very close to a 6850) at 1920x1080, with near-max detail, so I think you're fine at that resolution with a 6850.

Keep in mind I am saying the 6850 will be capable of providing approximately 60 FPS at that resolution in the games you listed. Whether or not your CPU will choke that performance to death sometimes (BF3 multiplayer, Skyrim) is another matter, but we can't really do anything about that.

Would I get better performance in the meantime if I threw in my old GeForce 6800 over my integrated HD 3200? Or would it be worse?

Your CPU will bottleneck any modern GPU in certain situations, but that's ok!! Just get the best video card you can afford.

Its OK if your CPU bottlenecks you, its still a huge improvement over your current GPU! Most of us are only worried about bottleneck because we are going from a 5xxx card to a 7xxx card or whatever and don't want to waste money for nothing. Enjoy your new GPU whatever you decided on!  

Oh crap, I'm sorry! I accidently posted that I had a 9850 CPU when it's really a 9600! Will this effect anything we were talking about? Should I still get the same card?  

No, wouldn't make much a difference.