This is a common misperception, actually, because we're taught that, like on an iPod, the iPad app update tool is how you keep all the software on your iPad updated. I mean, that's what it's called, right? :-)

Problem is, as you've found out, you can't actually update the operating system itself, what Apple is calling iOS, the OS for the i-devices, from within the App Store application. To update the firmware, you'll instead need to actually hook your iPad up to a computer and use iTunes. I'm suspecting that you already have a pair relationship between your iPad and some computer or other, whether it's a Mac or PC, but if not, you'll need to establish one.

Once you have the iPad hooked up to a computer running iTunes where the two are already paired, iTunes will actually go out to the iPad servers on the Internet and find out that...

As it says, you'll want to click "Update" to continue, at which point it'll warn you about this, that and the other:

Great, now it's time for a tiny bit of marketing, when Apple explains why this update is so darn cool and why you want to download it:

Still with us? Great. Click on "Next" and it'll start downloading the new 551MB software update:

Hopefully you have a reasonably fast connection: it took me about five minutes to download the update. While it's downloading, if you go to check what's happening, you'll see that iTunes knows something's going on...

Once it was downloaded, iTunes started to step me through the process it went through updating the iPad firmware:

And then...

And another step...

Finally, once the update is applied the iPad will restart and when it reappears in iTunes, it'll be showing the new version of the iOS firmware is installed and running:

That's the basic process. Once you've done this, yeah, why not go to App Store and see if there are updates to the apps too!