I agree with regards to the gtx610, it's just a waste of money.
The FX6300 is a good suggestion, it's a very good value, in terms of baseline performance an FX6300 system feels like an i5-3470, does better in some things, worse in others, but pretty much offers the same experience. And the FX6300 has very good overclocking potential and 95 TDP, so you don't have to go crazy on a CPU cooler. In fact, the new H60 2013 edition will be enough for a decent overclock on the FX6300, it doesn't run that hot, it's definitely a lot cooler than the FX8350. I would stay away from older H60's though, they are not good. I do prefer air coolers over prefilled systems because they offer much better value and long term reliability. For the price of an H60, you can get a top-tier tower cooler that will be quieter and will have better cooling performance for a bit more overclocking headroom. With a 20 % overclock, the FX6300 sits at the level of the i5-3570 at stock speeds in terms of allround computing experience, for only a third of the price, because even a cheap Gigabyte GA-970A-DS3 mobo can perfectly overclock an FX6300, and has all the useful features of a 130+ USD z77 board, ande is also an Ultra Durable glassfiber PCB with Japanese high precision capacitors, BIOS fallback options, etc... The only thing it misses is a VRM MOS-FET heatsink, but that's a very easy fix, and if you get MOS-FET heatsinks from an electronics store, it's a very cheap fix also.
The minimum graphics card I think makes sense to invest in is the 7850 2GB with 2 fans. Just because it's 4k capable (which is coming rapidly, 4k monitors are starting to pop up for just over 1000 USD), it has good gaming performance in 1080p, it's not loud with two fans, it runs cool with two fans, even with the maximum default overclock, has a 256-bit memory bus, DDR5 memory with a base clock of 1450 MHz, and the default overclock brings the GPU to a clock speed of 1050 MHz, all of that while only needing one 6-pin PCI-E connector.
A system like that, with a 970 AMD chipset mobo, an AMD FX6300, a cooler like the Scythe Grand Kama Cross Rev B for instance, or the Arctic Cooling Freezer A30, 8 GB of dual channel 2133 MHz RAM, an AMD Radeon HD 7850 2GB with 2 fans, and a nice case like the Fractal Designs Arc Midi R2, the Corsair R200 or if you want to go all enthusiast gaming style, something like a Cooler Master HAF-XM, is a very good mainstream gaming system that's for most users better value than a comparable Core i5 system, and it has the AMD benefit of being overclock-friendly, which is an ethusiast-experience bonus without the enthusiast pricetag.