It is obvious that I don't understand a few things. Basically, following Deadpool's suggestions I should get 4 cores, 8 threads. Either an i5-6500 (or possibly 6600K) or an i7-7700K and be done with it. OR: wait for AMD's 4C/8T lineup and take it from there.
Both of these options (maybe not the overclockable CPUs) are within my budget. I have a few questions though:
1) What is more important? 4 cores at higher clock speeds or 8 cores at lower/medium clock speeds? How much work threads do?
2) VRAM - 4GB, 6GB, or 8GB? Why? What is the best bang for buck?
EDIT: The rig is mainly for gaming.
8 threads are not necessary. You do need at least 4 physical cores (unless you can find a dual-core that´s powerful enough, which you won´t). HT affects gaming very little as of now. Keep in mind you have the new Intel architecture (it´s almost the same, with some minor improvements, +5%), so instead of a 6600K you could get a 7600k for around the same cash, and get something a little bit better.
The build in post #7 should be a good starting point.
About the cards, VRAM doesn´t affect how many FPS you´ll get. It is a limiting factor sometimes when there are very big and high res maps. For example, I think it´s the last Batman game, if you wanted to activate an option about the view distance and quality you needed at least 3Gb VRAM. Also if you have a high res monitor, you´ll need more VRAM, because you´ll be rendering many more pixels, and thus using more VRAM. There are a number of other specs that afect FPS, but VRAM is not one of them.
In your budget I would recommend the RX480, depending on your monitor, the 4Gb or 8Gb version.
Edit: You could get a 1070 if you are willing to spend a bit more.