To be put as bluntly as it can I think, right now you can not
run the game. Opening your case would help a lot as we could tell if you have a PCI, AGP, or PCI-e Express video card slot. That will determine what kind of card you should buy if you do want to upgrade. The rest of your computer is up to snuff and it seems like only the video card needs a slight upgrade. (I think my video card has SM3.0 but I'm not 100% sure and will get back to you when I figure out, as the card I have will be fairly cheap these days I'm thinking.)
As to the Catalyst thing you linked earlier, that is the graphics driver for the video card. I know that it (when downloaded in full) will put lots of icons and such on your desktop, however there is a way to install the latest driver without the extra things you don't really need (advanced user settings and such).
Don't rush into buying a card though. At the rate that new ones come out, buying a slightly better one might just leave you behind in a year again. Once/if you get a new video card, I or anyone here would be able to refer you to the proper video card driver.