Suggestions for new gaming PC? (Giveaway included)

I would go with AMD cpu if your budget is tight if not go with intel as for the GPU I would go with one of amds R9 or R7 cards.EDIT check Slickdeals for any good deals.Also amd Cpus tend to get hot so the stock cooler will not do.

I'm slowly building my new PC and I went with AMD cpu because I don't have a lot of money. I recommend checking out tom's hardware. there are a lot of good articles about different components.

CPU: Intel is the way to go (AMd only if you're tight on money)

GPU: Like already suggested AMDs R9 or R7. From Nvidia gtx 750/i

Im not sure what country you live... so prices could be different.
CPU:
The AMD FX 8350 would be nice since it's an 8 core and delivers good performance. The price in the netherlands is about 170 euro's.
Motherboard:
Get a simple cheap motherboard... but consider one which supports Crossfire/SLI for later upgrades. price range would be between 50 and 100 euro's.
GPU:
I would recommend the AMD R9 270x, which is available for 170,- And if you could add a 2nd one if the motherboard supports it.
You could consider the Nvidia 750 Ti which is a little cheaper and power efficient, but doesn't support SLI for later upgrades.
RAM:
I would say 4GB RAM @ 1333mhz... they are really cheap to get. Get just one 4 GB RAM stick and you could add another one later on. Its enough for gaming for a few years. You can get one for about 40 a 50 euro's.
PSU:
Get a Power supply unit with a least 550 Watt, Maybe even 650 Watt for later upgrades.
Also make sure you pick a PSU of a well known brand. 65 to 85 euro's.

The rest i think you can choose for yourself...
I would like to recommend getting custom cooler for your CPU.

The i7 series is better spent on i5. It really doesn't help in gaming.

Normally AMD Motherboards have more features for less money than one for intel. For 550€ you can get a Radeon 7870 on sale, or a R9 270x, with an FX-6300 you can fit it all these for 500-550€, I won't recoment you buying an Nvidia GPU because current AMD cards give more performance imo than NVidia's, if you want to spend up to 700€, get an FX-8320 or FX-8350 CPU, or and i5-3570K, i5-4670K, (or without the K if won't overclock), don't buy an i7, for games there is no difference.

My PC is full AMD with FX-8350 and Radeon 7950, cost me like 750€ my suggestion is if you are buying a PC from 500-600€ get an FX-6300 for sure, if more then think about i5 or FX-8320 or 8350, the best for the budget will be an FX. For PSU buy an 550W Corsair for like 45€, Kingston 2x4Gb 1600Mhz Ram, for CPU cooler a Cooler Master Hyper TX3 EVO, do not buy a DVD burner it's a waste of money, for MOBO that one that Snowtheory said is good, 1TB HDD from Seagate 7200rpm, for 500-700€ is better not buying an SSD.

My CPU and GPU are both from AMD, they have the highest cost/price ratio. If you lived in the US I would recommend Microcenter, their CPU and Motherboard bundles are great. Saved like 50 bucks. I know you don't judging by the euro sign, but I'm throwing that out there to anyone who lives in the US and is interested. I also went with AMD because I don't particularly like Nvidia and Intel as companies. Nvidia is being really cheap with Physx, and Intel has also been in some trouble trying to crush AMD. Monopolies are not cool.

In Holland we have a big hardware website who create a Best Buy Guide every 2 months. Basically Its a bunch of professionals that give you tips on best-bang-for-bucks, in short: whats the best to buy with the budget you have.
2 Years ago I created my new pc based on their guide, never been more happy with a gaming system.
Here's the system they compiled for 768 euros. Its in Dutch but the hardware speaks for itself.
Imo you can leave out the processor cooler (Scythe), which will take you into the budget you have. Intel stock processor coolers are good enough if youre not into overclocking.
Good luck!