Your initial comparison questions are pretty meaningless because they don't take into account motherboard costs, potential cooling, willingness to overclock, etc. Comparing APUs to Intel's core lineup is also pretty bizarre because they're totally different segments of the market. And totally different in price!!
Anyway the very basic kind of things goes like this (all CPUs increased in price to account for an appropriate motherboard):
An AMD A6 APU (~$135 for CPU/mobo/stock cooling/integrated graphics) is excellent for an entry level consumer/office computer. It'll be fast, peppy, have a bit of graphics oomph in a pinch, and basically do easy stuff well for cheap.
Intel competes in that segment with a dual-core Pentium like the G1610 (~$110 for CPU/mobo/stock cooling/integrated graphics). A little cheaper, weaker in terms of graphics for light gaming, fewer cores but a bit faster on each core. Again, good at casual computing for cheap.
The next tier of price looks at an AMD A8/A10 APU vs an Intel Pentium with a discrete video card, for entry-level gaming.
An AMD A8-A10 (~$210-240 for CPU/mobo/stock cooling/stronger integrated graphics) is a good all-around entry-level gamer. There are few games that won't run, but many where you'll have to turn various settings to low.
An Intel G1610 with the additional cost of, say an AMD 7770 (~$220 for CPU/mobo/stock cooling/discrete graphics card) is also an excellent entry-level gamer. It'll be stronger for many games than the AMD A8/A10 in terms of raw graphics power and core speed, but it'll also be weaker for some popular games that like a quad-core rather than dual-core CPU.
Then we go up to more expensive, more mid-level competition (Intel quad-cores, AMD CPUs for use with discrete graphics card). I'll be cutting of consideration of graphics here, as it's not needed for comparison - each CPU should be assumed to be paired with an equally-strong graphics card. I'll be assuming you plan to overclock the AMD CPUs.
Now we're looking at an overclocked AMD fx-6300 +appropriate mobo + budget cooler (~$260, but again, no graphics in that amount)
versus a locked (un-overclockable) Intel i5, like an i5-4570 (~$270 for CPU/mobo/stock cooler)
Performance between those two is roughly equal, both being quite good but not quite perfect for gaming. Going farther up you can get into overclockable i5s, which is actually a much bigger improvement than getting a usually-pointless i7 (difference between desktop i5 & i7 is just one rarely-used feature pointless for the average casual user). There are much bigger concerns for professional users who need computing power to make a living, but we don't need to go there.
That should be a MUCH more relevant series of comparisons compared to what you started with. You'll see pretty quickly that price/performance is pretty even across most budget segments right now once you take motherboard costs into consideration and know which CPUs represent what part of the market.