You need to get the latest BIOS from Dell. A11 I think. There are people running all kinds of modern GPUs in that.
http://www.userbenchmark.com/System/Dell-OptiPlex-GX620/2140
I'm going to assume you have the Tower version.The others need a low rpofile GPU, and there are limited PSU upgrades available.
The fastest PentiumD 960 barely equals the slowest Core2 CPUs.
Any DDR5 GPU will be better than DDR3. But I assume money is a factor here.
The 3GB RAM will be an issue. But one that can be resolved. Physically you can install 8GB RAM in that computer. The BIOS will only use 3.5GB, and so will Windows 32 bit OS. A little bit more will be used by the chipset. The trick is to go ahead and install 6GB DDR2-667 RAM. The 2GB video card will...