I am going to assume you have PCI-2.0 as this CPU i believe is about 6-years old. 600w power-supply should be solid for most single card setups. You have a number of options to take a look at. What size monitor are you using? What is your price range? 
 
Cheapest route with a significant upgrade in speed over your older card would be a 1060 -now this comes in two flavors, if you are using a 1080p monitor 6GB is more then you need while 3 gb is jsut under what i have seen used... so i tend to recommend going the 6gb. You can get this on Amazon as well. These run slightly better then my old titan GTX cards and my wife is running wow at just about 1080p(She has a 16:10 monitor 1600X1200 or something like this) @ max settings with mild AA at 40-100FPS. So expect similar performance.
 
http://www.evga.com/products/productlist.aspx?type=0&family=GeForce+10+Series+Family&chipset=GTX+1060+6GB
 
If you want to spend a bit more money, the 1070s are fantastic but maybe overkill for what you need. They are a great card for the value and my #1 recommendation for you. Here you will prob start to see your system holding your card back rather then the other way around.
 
http://www.evga.com/products/productlist.aspx?type=0&family=GeForce+10+Series+Family&chipset=GTX+1070
 
You could go the 1080 route, but i do not feel that would be worthwhile on your older hardware.
 
In all cases above your PCIE 2.0 is going to be just fine (only a few less % performance then PCIe 3.0)
 
If it was me, and I wanted a couple more years, I would opt for the 1070GTX. Make sure you check your reviews though and confirm the frame rates at the resolution you are playing with are acceptable to you. 
 
Best,
 
Jonathan

Her is the link. 
 

A 1050 or 1060 should work perfectly with the motherboard in your system. You may want to consider going with this gigabyte 1060 as it is the only 1060 confirmed to be working in your pc. A evga 1050/1060 should work in your system without any issues too, but I couldn't find any confirmed ones using the userbenchmark website, so I thought I would suggest a confirmed working gpu to prevent any compatibility issues.
 
All userbenchmark results for the gateway fx6800-01e system can be seen here.
 
Note: Make sure your motherboard bios is completely up to date before installing any newer gpu's into your system.
 

RN@VGA

Sajin
A 1050 or 1060 should work perfectly with the motherboard in your system. You may want to consider going with this gigabyte 1060 as it is the only 1060 confirmed to be working in your pc. A evga 1050/1060 should work in your system without any issues too, but I couldn't find any confirmed ones using the userbenchmark website, so I thought I would suggest a confirmed working gpu to prevent any compatibility issues.
 
All userbenchmark results for the gateway fx6800-01e system can be seen here.
 
Note: Make sure your motherboard bios is completely up to date before installing any newer gpu's into your system.
 

So I recently purchased a EVGA gtx 1060 SC 6gb for my gigabyte GA-Z68XP-UD5, and it works good so far. 
 
However I was thinking of trying out the EVGA gtx 1070 SC, but since my mobo doesn't have UEFI, then I don't know if it will work. They're both EVGA cards, but I don't know if each model varies. 
 
Is there a way to check? Maybe you know more about this.

All evga 1060's & 1070's use a uefi vbios. Since the 1060 works fine in your pc there is a good change the 1070 will also work. The userbenchmark website shows 2 confirmed working 1070's on your motherboard, none of them are evga 1070's.