As a new user of Ubuntu you should not have to download anything from anywhere to get almost all running smoothly without issues.
This is especially true for graphics. If there was a bug or any other incompatibility in such brand new and partly untested downloaded drivers they will let you alone with a system booting to a command line terminal only, or in the worst case with a system that can not boot at all. Then it may not be so easy to get rid of these drivers again.
There is a great effort from our community to provide you with almost all possible drivers to make sure your system boots. Amongst these the Open Source ATI driver
"radeon" is very stable, and it is well supported by ATI (ATI engineers develop this driver for us).
Only very new graphic cards, or applications demanding a lot of GPU power may benefit from a newer, proprietary or untested driver. Only if we had issues with our present setup we may consider installing a newer proprietary graphics driver.
This is not the usual case.
Many people are very happy with the default drivers installed from the installation CD. They would not have much if any benefit from a newer driver.
In your case after you had already seen that your computer runs well in the live system I recommend you just install Ubuntu and enjoy it as it is. Refrain from updating or upgrading things with downloads from the Internet.
This will very likely break your system sooner or later.
In case you have graphical issues - look around for other people with the same problem to see how they solved it. Then you may consider installing a proprietary driver which is only a mouse click away.
Also see: Is my ATI graphics card supported in Ubuntu?