But my question is, can I just throw in a new graphics card or am I going to need to upgrade mobo, ram, or processor as well to accommodate it?
To answer this question: No, you don't need to upgrade anything if you replace the graphics card. You will need to uninstall the GPU. And uninstall the driver software. Make sure you download the most recent drivers for you new GPU prior to making the switch.
And if you stick with AMD, make sure you get a decent price (they're currently in high demand because of bitcoin mining, so read some reviews to find out what the original MSRP pricing is). For example, the r9 280X had an original MSRP of $300 but miners have moved the purchase price to ~$400. I'm sorry, but I'm not paying a premium for a GPU just because miners want it.
While it's long in the tooth the PII x4 965 is still a decent CPU. If you would like to try overclocking, search the internet using your CPU and motherboard in the terms. There are several youtube videos that provide a pretty good demonstration of how to do it. If you do want to OC, I would suggest getting a better CPU cooler. There's several Artic CPU coolers that are pretty cheap and are much better than the stock one.
My only other suggestion is maybe trying an SSD. While it won't improve in-game play, per se, it will add new pep to your system.
Hope this helps.