If you're not so hot with the technical side of PCs I'd recommend you don't rush into buying anything without spending a few days reading forums, a new graphics card would definitely help Skyrim in your case but I'd probably go for Nvidia instead, Bethesda's engines have traditionally worked better on them.
Both the PC game and the Xbox360 game had been developed using nothing but Radeon 9800 XT graphics cards in the Bethesda shops, until near the end, when they got in several Radeon X800 XTs.
ATI apparently really did make tests, and did not name their 9600 series as being useful. They seem to have made little effort to be certain that anyone knew that the 9600 had a bigger number than the 9500, simply because it was newer, not that it was nearly as good. They named the 9500, skipped over the 9600, and named the 9700 / 9800 as good, and all of the next generation's upper end from the X700 on up.
Some moron at Bethesda added the 9600 in before they published their requirements, which were total crap overall because of the morons believing nVIDIA, mostly.
