Couple of things.
With that budget, an integrated GPU is well below what what your budget will allow. APU systems (the ones that just use the GPU on the procssor) are nice for 'every day' machines, but for serious gaming you'll be going the dGPU route (separate graphics card) to begin with.
With myself being an unabashed AMD fanboy, I'll still recommend an Intel i5 or i7 system. It looks like new Intel processors are scheduled for release in Q2 2014, along with the new Z-97 chipsets, so if you want to stay 'ahead of the curve' you'd likely want to pick up one of those chipsets. You can always upgrade the CPU later if you decide to with a lower end model. That being said, Intel i5 and i7 processors do command a bit of a premium, due to the marked advantage they have against current AMD standalone CPUs. You could probably make do very well with a low end i5 if you think you will have more money to upgrade to an i7 later. Just keep in mind that Intel will go to a new processor socket at some point, so you will eventually get left behind.
As for graphics, that can change quarter to quarter, or even month to month, as to which graphics card is the best buy. Really you can't go wrong with either NVidia or AMD, but the latest higher end AMD R7 and R9 GPUs using the GCN 1.1 architecture will often (depending on the model) include TrueAudio as part of the package. TrueAudio is brand spankin' new technology that supposedly is a welcome improvement r.e. sound positioning technology, but so far no games support it. Thief, which is scheduled for release later this month, will be the first official title to use TrueAudio technology.
AMD GCN 1.1 GPUs are also Mantle Ready. Mantle is the latest innovation from AMD, which bypasses DirectX to a large degree to allow for more CPU overhead. That being said, with the processor that your budget SHOULD allow you to get, you probably won't be running into many CPU limited situations, at least not currently. Future games may require a lot more CPU power if this 'close to metal' thing takes off. Microsoft announced that they are looking at implementing their own 'close to metal' solution for DirectX just a few days ago.
Again, really you can't go wrong either way, and there are rabid fans of both Nvidia and AMD. I'd say go with the best deal on current gen GPUs that fit in your budget. If you are planning on playing any games using the Nitrous engine (from that new company Frogboy is part of), you'll probably want to look hard at a R9 290 or R9 290x... Oxide is all about supporting Mantle at the moment! Otherwise, look at the benchmarks of the games you'll be expecting to be playing regularly and see which card is in your price range, that does the best in the FPS and quality categories. See links below for a place to start r.e. reviews on the various cards.
As for memory, these days 8 GB minimum, 16 GB is happier. Faster memory with lower timings is helpful, but not absolutely essential - the gains to be had here are small percentage point bumps in performance, so more memory is better than lesser but faster memory. While you can buy two larger sticks now and two larger sticks later if you are looking to upgrade, keep in mind that mixing different models/brands of memory sticks is not good practice, and can cause stability issues. If you are looking to double your memory at some point in the future, I'd recommend making that sooner rather than later, otherwise you may end up tossing your old memory sticks and buying new ones to go with your new pair...
So, in short, if you don't need a new computer immediately (i.e. can hold off until spring/early summer), I'd recommend doing some research between now and then. Check out the reviews on the various tech sites r.e. how well the hardware you are thinking about does in benchmarks and such. Two sites I read daily are:
www.guru3d.com is also a good site to check. You can see what THEIR experience with various setups is, to help you arrive at your own decision.
Finally, a good strong power supply is essential. Don't scrimp here or you may regret it later. Overkill isn't necessary unless you have money burning into your pocket, but pay close attention to the recommendations you see in their 'build recommendations' articles. Various review/tech sites will usually put together a buyer's guide/laundry list of stuff at various price ranges. Here's a link to a $1325 system that Toms recently put together for one of their guides:
Something to keep in mind if you are looking for a 'deal' is that you may see some price drops on the Z87 boards when the new Z97 boards become mainstream this spring/summer. So if you don't mind being just a little behind the curve you can still get some killer performance with the current generation of Intel boards. You will get left behind eventually though, so consider how long you want to stay with the current system. Just keep in mind that the motherboard socket determines which processor you can use, so the socket and processor you buy now may be obsolete soon, as both AMD and Intel like to migrate to new CPU sockets every couple of years it seems.
The Z97's have some newer features, though, so take a look at those to see if don't mind doing without them before going the 'closeout bargain' route.
DDR4 memory support for consumer/enthusiast level boards is supposed to be coming soon. DDR4 should be faster than DDR3 of course, but to be honest, a good high end GPU will give you MUCH more performance in gaming than faster system memory. I wouldn't hold out for DDR4 unless you don't mind waiting for a while, as DDR4 will be more expensive to buy. Intel is looking at Q3 2014 for DDR4 support, based on some recent articles. You'll need a DDR4 mobo to use DDR4 memory.
Oh, if you go the Win 8.1 route (likely), make sure you get Start8 from Stardock! (unabashed plug).
Finally, as mentioned above by others, a good SSD drive is essential for high end performance. That being said, a 128 GB should be plenty for starters, with a 1 TB HDD (preferably 7200 RPM) to go along with it. You can always upgrade to a larger SSD later, when the prices are lower, at least that's been the trend year on year so far -SSDs getting cheaper every year. Or you install a second SSD instead of just replacing your current one...
One other note...
'Ready Built' machines often include some useful software, such as Microsoft Office and such. If you build your own machine, you WILL need to buy Windows and these add ons separately. That being said, there are some boutique builders that make enthusiast level machines, such as CyberpowerPC, Xoticpc.com, etc.. They won't have as much bloatware as say your average HP machine, but they do have 'Microsoft Office' packages they can bundle with your system, for a fee of course!
Buying off the shelf at BestBuy, WalMart, or wherever will get you a 'canned' system, that will likely be a bit behind the curve. You will have a warranty, though... personally I get more enjoyment out of building systems, or at least I did until I went the laptop only route...