5770 or 5830 or 5850

The HD5770 will be pretty good for 1920x1080. You wont be able to max out the settings on the more intensive games but it will still be a very capable card. If you can't afford the HD5850 then perhaps an HD5770 for now and another in crossfire when you can afford it(if your motherboard is crossfire.)
I would forget about the HD5830. It's rather overpriced for it's performance IMO.