My "mid-range system" is a Haswell non-K i5 ($180) on an H81 or B85 ($55-70 depending on if OP is ok with having only 2 RAM slots), a 270x or equivalent ($200), 2x 4GB DDR3 ($80), ~450w good brand power supply (~$40), 1-2TB hard drive ($60-80) OR 240GB SSD ($120), cheap case ~$50, and optional optical drive ($20).
Costs are around $650 to $750, depending on choices. OP can step down to an i3 and shave ~$50 off, but that $50 goes a long way. Downgrading to a 2GB HD7850 at $120 will lose about 20% GPU performance but shave off $80. After either of those, moving down to a single 4GB DIMM is an option to cut out another $40.
The "upper low-end" PC with an i3, 7850 and 4GB of RAM runs just under $500. This is basically the PC I built for my wife. A 7850 is about as fast as the GPU in the PS4, and should give a pretty good gaming experience for a while.
How I'd pair processors to video cards for gaming:
Pentium/Athlon -> HD7750 - 7850
i3 -> HD7790/R260x - HD79xx/R280
i5 non-K -> 7850/R265 - R290
i5 K -> 7870/R270 - 290x
i7 -> 290+ / dual GPU configs
Insert nVidia equivalents.
I tend to prefer spending a little more on a processor than most on this forum, but I do more with my PC than game, and CPUs seem to lose value less quickly than video cards.