You can find out what kernel version you are currently using by issuing the command
Which will return the kernel version for you as seen below.
Linux compost 2.6.5-1-386 #2 Fri Apr 30 20:13:30 EST 2004 i586 GNU/Linux
You can upgrade the kernel like any other package using apt-get but you will need to re-boot and likely modify your boot loader. The only querk of the kernel package is that you don't ,generally, select a specific kernel. Instead you select a dummy package that depends on the kernel version you want. For instance you select the 2.6 dummy which currently will istall the actually kernel version 2.6.5-1. In this manner you always track the latest version of the 2.6 series kernels. If you are using lilo you must run "lilo" after you make changes to "/etc/lilo.conf" or there is a very good chance your machine will not boot.
In the case of the upgrade from woody to testing (again at the time of writting) it was necessary to add a line to lilo.conf to allow the kernel to use an initial ram disk. This was done by adding the line "initrd=/initrd.img" to the "image=/vmlinuz" stanza which produced a stanza looking like this
image=/vmlinuz label=Linux read-only initrd=/initrd.img # restricted # alias=1
You did remember to run "lilo" didn't you! Time for a re-boot then. This is the make or break bit of the job. If you have screwed it up your machine won't boot. To be quite honest though its pretty hard to screw it up but bad things can happen!