This one will serve all your slackware needs at the start. http://slackbook.org/html/network-configuration.html
Take a day or a weekend and just read the whole book. Not too much, not too deep just the right proportion to get you going.
For a quick help. Do you have dhcp active on your router? Try a dhclient eth0
on the console.
What exactly do you mean with "running on a desktop". A desktop computer or a desktop installed (like KDE, Gnome).