I currently have a so-so gaming PC, but I want to upgrade it to run modern games such as Planetside 2, Crysis 3, BO2, and future games on at least Medium settings, preferably high. I'm willing to spend $400, but would like to stay in the $250 range.
GPU - ATI Radeon HD 4800 Series
CPU - Pentium(R) Dual-Core CPU E5400 @ 2.70GHz, 2700 Mhz, 2 Core(s), 2 Logical Processor(s)
Raidmax makes PSU-shaped objects.
OP, if your PSU is actually capable of 500W, i.e. it isn't some cheap junk that came with your case (e.g. Raidmax, Apevia, Logisys, Diablotek), then a HD7870 for around $220 would do quite well for you. Your CPU is weak too, but not hopeless. Your next upgrade can be CPU/Mobo/RAM; something like an i3 on a H77 with 8GB (or even just 4GB) of RAM.
If your PSU is not up to the job, you can get one with the money left in your budget. Get something around 550W made by Seasonic, Superflower, FSP, Delta, or Enermax/LEPA. Brands made by one of those include Seasonic, some Corsair, Antec, new Rosewill, and XFX.
If your PSU is of decent quality, it will support a graphics card as strong as a GTX670 or 7950.
A GTX670 is a $400 card, but I might suggest that a GTX660 @$230 or so would be a significant boost in graphics capability that would meet your major objective of running at high settings.
Similarly, a 7850 0r 7950 if you prefer amd.
You might be concerned about the ability of your older E5400 to cope.
Most games do not make good use of more than 2 cores, so you may be ok.
To help clarify your options, run these two tests:
a) Run your games, but lower your resolution and eye candy.
If your FPS increases, it indicates that your cpu is strong enough to drive a better graphics configuration.
If your FPS stays the same, you are likely cpu limited.
b) Limit your cpu, either by reducing the OC, or, in windows power management, limit the maximum cpu% to something like 50%.
This will simulate what a lack of cpu power will do.
Go to control panel/power options/change plan settings/change advanced power settings/processor power management/maximum processor state/
set to 50% and see how you do.
If your FPS drops significantly, it is an indicator that your cpu is the limiting factor, and a cpu upgrade is in order.
It is possible that both tests are positive, indicating that you have a well balanced system, and both cpu and gpu need to be upgraded to get better gaming FPS.
If you decide you want to upgrade the cpu, you are looking at new ram and motherboard as well.
A ivy bridge cpu is likely the best way to go. A i3-3220 is about $130, or a i5 quad is $200. A Z77 motherboard will be <$100, and since ram is cheap, I would suggest a 8gb kit for about $45. If you current copy of windows is oem, you might need a new os too.
I'd stay away from the i3 cpu for now, even Tom's isn't recommending anything other than a 4 core cpu for most price points now. Ideally you'd be able to get the fx6300 cpu and a Hyper 212 Evo cpu cooler, but that's just outside of your budget.
Like Onus said though, you could spend the 250 dollars on a decent PSU and get the 7870 now. Sapphire has one for 225 or so, and there is an Antec 520w PSU for 40 after all MiR's are done. Those 2 parts would provide a solid upgrade, and a good basis for replacing the motherboard/cpu/ram in the nearer future.
I'm sorry TESKA did you say something - after giving poor advice about power supplies, I'm afraid (hope) most people will not listen to you. The big question is - what 500W psu does the OP have. The GTX660 and HD7850/70 are good suggestions. Onus ans geofelt have been giving good advice for a long time - you can trust them.