Vram isnt what you should be looking at. It is not the number that means anything towards the performance of the card.
To a degree, the core clock is one of the best ways to see how good a card is. However even that doesn't tell you much.
I'll just say this, with a gtx 560 you can max out the game and it is incredibly smooth.
5770 is a bit weak. For one, amd has not yet implemented the full dx 11 codepath like nvidia has, so the card struggles to compete against a comparable nvidia card. Secondly, it's too slow if you buy the equivalent nvidia card (say gtx 550) unless you have a pretty low resolution. Generally speaking, I would shoot for:
gtx 460 768 on 1440x900 or lower
gtx 460 1gb on 1680x1050 or 1920x1200
gtx 560 ti for anything greater than that
Also, as mentioned earlier, civ5 is cpu limited in many instances rather than gpu limited. Generally speaking, a 3.5 ghz + quad core should be sufficient on most maps/settings.
I play at 1440*900. The graphics are set to lowest because currently I don't have any graphic card. I won't change the processor, however RAM was cheap & I think 3GB RAM is too less for games.What resolution do you game at? You i3 is probably more cause for concern than only 3gb of ram.
might want to look into a used ati 4850 or 4870. both can be had for under $100 and still strong cards just not DX11 only capable of DX10. Off the wall thought there ;-)
I wouldn't reccomend the 560ti, it's a weaker card overall than the 460. The 570 is more powerful and expensive though. Don't know much about Radeons can't speak for them.
If you're in Australia you can buy from a site called PC Case Gear for REALLY cheap.