
Check file permissions and directory permissions to be sure that access is neither too restricted nor too easy. I find that Slackware tends to lean toward a more open environment than I like, so I go around changing 755's to 711's for binaries in the .../bin directories and stuff like that. Or even 700's in the .../sbin ones. Especial care is needed if you've carried over ftp, telnet or web servers; but then, if you were running those, you probably thought of that already. :)
Look at /etc/inetd.conf or /etc/xinetd.conf and make sure you're not running any Internet services you don't need to. Also go through the boot scripts in /etc/rc.d and friends for the same purpose. Check your firewall rules if your box is an Internet gateway or has Internet access.