For gaming, intel of some sort is the way to go.
AMD cores are slow. The best might have a single thread passmark rating around 1500.
The i3-6100 mentioned above is 2102.
Here is a post I made earlier regarding a "budget" build.
Prices will have changed. Take from it what you will.
------------------------------ budget skylake build ---------------------------
For a budget gaming build, I like to recommend that one builds for future expandability.
That means paying a bit more up front for some parts that allow for an easier future upgrade.
A good rule of thumb is to budget twice as much for the graphics card than for the cpu.
Let me start where you might not expect:
1. Buy a good 500w psu or better. 500w will run a card as good as a GTX970. 600w will run a GTX980ti.
Future graphics cards will be built on smaller 14nm so they should not need more power than today.
I would normally suggest Seasonic 620w: Expect to pay around $65.
Look for a tier 1 or 2 quality unit on a list such as this:
https://community.newegg.com/eggxpert/computer_hardware/f/135081/t/45344.aspx?Redirected=true
2. Buy a Z170 based motherboard. Z170 will allow you to install a overclockable cpu and even permit a future Kaby lake processor upgrade.
You should find one for under $100. Lesser lga1151 chipsets will work for the truly budget constrained but at the expense of future upgrade optiions.
Here is a M-ATX example: ASRock Z170M Pro4S for $99:
http://www.realhardtechx.com/index_archivos/Page362.htm
3. I suggest a I3-6100 @3.7. About $130. If you are truly budget limited, you can buy a g4400 dual core for $60.
In time, you can upgrade to any I3/I5/I7 cpu that you want and market theoriginal processor.
4. The intel stock cooler will do the job. Skylake runs cool.
If you want, you can use a cryorig H7 cooler with a 120mm fan. $35. It will be quieter.
http://www.newegg.com/Product/Product.aspx?Item=9SIA4UF2DZ6565
One can always add a cpu cooler later.
5. For ram, speed is not important. Buy a 8gb kit of 2 x 4gb DDR4 1.2v 2133 speed ram.
About $40 for this:
http://www.newegg.com/Product/Product.aspx?Item=N82E16820148858
Heat spreaders are marketing and generally useless.
Faster ram is not worth it for skylake:
http://www.silentpcreview.com/article1478-page1.html
6. Cases are a personal thing. Buy one you love. Most will do the job for <$50.
Here is a silverstone PS08 for $35:
http://www.newegg.com/Product/Product.aspx?Item=N82E16811163223
7. The graphics card is the most important component for gaming.
I like the GTX750ti and EVGA as a brand.
Here is a superclock version:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
You could go stronger in the video card if your budget permits and your games need it.
On the other hand, you could build using the integrated 530 graphics which is quite good and see how you do.
By deferring on the graphics card, you will get a better idea of what you really need.
Integrated is fine for sims, but not fast action games.
8. Lastly, I will never build again without a SSD for the "C" drive. It makes everything you do so much quicker. 120gb will hold the OS and a handful of games.
With 240gb you may never need a hard drive at all. Defer on a hard drive until your ssd approaches 90% full.
Currently, I like samsung 850 EVO best.
-------------good luck------------
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