I'm running Debian Squeeze completely up to date on a machine. However when running
uname I see that I'm on a very old kernel:
$ uname -r 2.6.32-5-amd64
Is this normal for Debian Squeeze? Should I upgrade? If so, how?
Yes. 2.6.32-5 is the current version of the kernel in Debian Squeeze. You'll notice the linux-image-amd64 dummy package lists linux-image-2.6.32-5-amd64 as a dependency.
Squeeze is a old stable version. The new stable version, wheezy, is in "release candidate 1" state, so an old kernel is normal in squeeze.
I suggest upgrade to wheezy or activate "backports". Backports are new stuff ported to squeeze.