People who recommend i3 should buy and actually use it themselves to game on. With pictures/videos as proof etc. 😀
I used an i3-540 OC with a 470 and 7950 as well as 7950 CFX.
I used the same 470 (SLI) with a 555, 965, 1090T, i5-2500k as well.
For a budget gaming rig I'd consider an i3, it has it's pros and cons just as the AMD chips do. AMDs chips are good for highly threaded games, but still suffer from poor single thread performance in those games and is exemplified in MP gaming. Each side in going to have it's advantages when comparing the i3 to even the 8350, AMD can overclock, but never to make up the gap in their ST performance. The i3's can't overclock, but they have stout ST performance out of the box compared to AMD and have vastly superior perf/w characteristics.
For me personally it's very hard to recommend or consider an AMD chip for gaming, after owning so many recent chips with water cooling and high clocks they just don't have the the single thread performance to drive performance in the games I tend to play, which are typically RTS/Diablo type games/Online based.
However after using the now, extremely dated 2008 tech i3-540 overclocked to 4/4.2GHz I can also say it isn't free and clear of problems of it's own, it does not do well in some titles, like BF3 MP, in single player there is no problem, but it doesn't not excel in MP with a typical cap around 44-46 fps. It's also no longer the creme of the crop when it comes to IPC, and had a hard time in some of the titles that do favor IPC such as Grim Dawn.
In the end you need to look at the types of games you play and then decide between the two, because they both have strengths and weakness. Anything short of the i5 is a compromise in gaming, if you haven't already consider used... You should be able to find an i5-2500k or i5-3570k for possibly as low as $150.
As far as RAM goes don't over spend, it makes very little difference in gaming.