If the $250-$350 budget is all for the GPU, a GTX 660 ti at around $300
is perfect for maxing out current games at 1920x1080. If you are interested in overclocking, you might be able to squeeze a bit more performance on average with an HD 7950, which can also be had for $300
, but the 660 ti and HD 7950 are pretty close as far as I can tell.
You could probably do fine saving yourself $50 or a bit more going with an HD 7870, or even HD 7850, but those aren't guaranteed to "max everything out" at 1920x1080, though the HD 7870 would come close in most games. Even the HD 7850 wouldn't be so far off if you overclocked it. A 660 ti or HD 7950 would be more future proof though.Edit:
I second the ram upgrade considering how inexpensive it is these days. Other than that, you'd probably be looking on getting a new motherboard and CPU (which is probably not necessary at this point).