You want to back up as much as possible. The major exception is software that can be easily reinstalled, but even they may have configuration files that it is important to back up, lest you need to do all the work to configure them all over again. Another major exception is the /proc filesystem; since that only contains data that the kernel always generates automatically, it is never a good idea to back it up. Especially the /proc/kcore file is unnecessary, since it is just an image of your current physical memory; it's pretty large as well.
Gray areas include the news spool, log files, and many other things in /var. You must decide what you consider important.
The obvious things to back up are user files (/home) and system configuration files (/etc, but possibly other things scattered all over the filesystem).