"I wanted to upgrade the kernel, possibly to 2.4.24 ... or maybe 2.6. How would I go about this?"
Whenever you upgrade the kernel you should install the new kernel as a second kernel with its own initrd (if you use initrd). Set up your bootloader to dual boot the two kernels. You do not remove the original kernel until the new kernel has been working OK for a while (a week or more).
"How do I make sure all users of the system retain their settings and files?"
If all that you are doing is adding a new kernel then you will not touch the user settings and files.
"How do I update packages ... if I have to?"
I do not know how to run Slackware updates. But in any case when you update your kernel I recommend that you compile your own from source that you down load from www.kernel.org
"I'm going to have to do this sooner or later, so I might as well learn it now."
I suggest that you skip learning how to substitute a new Slackware kernel for an old one and proceed directly to compiling your own kernels.
Be prepared. Create a LifeBoat CD.