Currently I have:
 
GeForce GTS 250
AMD Phenom 9600 Quad Core Processor
 
And after running the Final Fantasy XIV benchmark I've decided that I want to update my computers innards. What does everyone recommend on the graphics card/processor side? I don't have $700 to spend so something between $200-$300 would be nice. Each, obviously. Not altogether.

" Currently I have:  GeForce GTS 250 AMD Phenom 9600 Quad Core Processor  And after running the Final Fantasy XIV benchmark I've decided that I want to update my computers innards. What does everyone recommend on the graphics card/processor side? I don't have $700 to spend so something between $200-$300 would be nice. Each, obviously. Not altogether. "

Are you on a laptop?

don't buy AMD, Intel is way better...

" Are you on a laptop? "

if your mobo supports it, get a Phenom II 
if not, either upgrade your mobo and get a Phenom II, OR get a new mobo and a Core i5 or core i7 if you've got the cash.  i recommend sticking with AMD or Core i5 so you dont have to buy triple channel ram. 
 
for graphics cards, maybe get a 5770, or a 5830, OOOOOR even a 5850 (that should SMOKE most new games, except for...well crysis and metro 2033)

Your cpu is quad at least, it's sort of slow. Just overclock it if you can. For the $300 option, maybe get a gtx465 videocard from nvidia. It might not be as fast as the gtx470, 480, however it's probally good enough. As stated for the above post, maybe a hd5830, hd5850 is also good choices, the hd5850 will be over $300, no phsyX. (re-edited)
 
I would check your mobo for what kind of amd cpu it can accept, like can it work with AM3 socket, then decide from there.

The computer is a Gateway FX7200. Not sure of the motherboard. :(

I would hold off on upgrading your hardware if this is just based on FFXIV's benchmark performance. It's obviously still not optimized (there aren't even any options), and the performance is not yet final. I'd suggest waiting.
 
However, if you insist, look at the Phenom II X4 955 or 965, both of which should work fine on an AM2+ board. For graphics cards, consider the HD 5850. 
Though, I would really recommend holding off if FFXIV is your only reason for wanting to upgrade and wait till you see how it is when it's released.
 
BTW, 47 meant the 5830 and 5850, not the 4800 series.

Phenom II 965 BE - $180 
Radeon HD 5870 - $400 
 
Alternatively, since FFXIV will be an Nvidia-optimized game you can go the Nvidia route with 
 
Athlon II 635 -  $100 
GTX 480 - $500 
 
The clock speed difference will lower CPU-bound performance a bit, but could be mitigated with some overclocking. The GPU performance would have a 15-20% edge over the ATI solution, but will be hotter and more expensive to run due to higher power expenditure.

" Phenom II 965 BE - $180 Radeon HD 5870 - $400  Alternatively, since FFXIV will be an Nvidia-optimized game you can go the Nvidia route with  Athlon II 635 -  $100 GTX 480 - $500  The clock speed difference will lower CPU-bound performance a bit, but could be mitigated with some overclocking. The GPU performance would have a 15-20% edge over the ATI solution, but will be hotter and more expensive to run due to higher power expenditure. "

 BTW, 47 meant the 5830 and 5850, not the 4800 series. "

" I would hold off on upgrading your hardware if this is just based on FFXIV's benchmark performance. It's obviously still not optimized (there aren't even any options), and the performance is not yet final. I'd suggest waiting.  However, if you insist, look at the Phenom II X4 955 or 965, both of which should work fine on an AM2+ board. For graphics cards, consider the HD 5850.  Though, I would really recommend holding off if FFXIV is your only reason for wanting to upgrade and wait till you see how it is when it's released.   BTW, 47 meant the 5830 and 5850, not the 4800 series. "

Would I notice much of a difference going from the Phenom 9600 to the i7-930? And would I need better cooling?