Best budget CPU? Help!

For gaming, you want fast cores.
For editing, you want many cores.

The FX-6300 might be good for editing but not for gaming.
It has a total passmark rating of 6344 and a single thread rating of 1408.
Your upgrade to a FX-8350 might bring the total up to 8949, but the single thread rating would remain at 1504.

The G3258 is an interesting option.
At stock 3.2 , the total rating is 3495 and the single core rating is 2175.
With overclocking, it can be 20% better.
Be advised, some games actually need 4 threads.
There are upgrades to i3.i5, and i7 on an older tech which is not bad.

Then there is the $60 G4400.
It turns out to be a surprisingly good gamer.
Here is a set of tests comparing the G4400 to a X4-860K. Both stock and overclocked.
http://www.hardwareunboxed.com/forum/viewtopic.php?f=13&t=1574

The nice thing is that with a lga1151 motherboard, you have all sorts of upgrades available; i3,i5,i7, and perhaps the future kaby lake.
The report includes BCLK overclocking, but I would not count on that.
Mo matter, the G4400 wins over the overclocked X4-860K virtually everywhere.

Now, if editing is your most important activity, and you use an app that is multithreaded a i3 or i5 is probably better.
To mu mind, it is better to have longer batch application runs than to suffer stuttering in games because of a slow cpu.

Define "best"

On a budget, a overclocked G3258 is certainly a reasonable choice.

You will be buying into an older generation of cpu and motherboard.
If you go with the cheapest motherboard that allows G3258 overclocking, it will not be suitable for a future cpu upgrade that might include a overclockable "K" processor.
You have not indicated how important your editing apps are to you and if they can take advantage of multithreading.
If they are single threaded, then the G3258 will be very good.

I always like to think of a budget build as one which can be grown.

Here is a discussion on that which I made some time ago.
Take from it what you will.
And remember that prices and products will have changed.

------------------------------ 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------------

That looks good to me.
You can hardly do better with that budget.
And, there is still some upgradeability.