Are you using Xfce or KDE? If so, what are you using to make the connection? That is, what software package are you using to manage your connections? Slackware doesn't come "stock" with such a program, but Wicd is available in the /extra directory on the installation CD or from a Internet mirror, e.g.,
http://ftp.osuosl.org/pub/slackware/
Install wicd and then reboot your computer to load the wicd module (driver). It will do it automatically each time thereafter. There is a command to load the module without having to reboot, but it doesn't come to mind...... OK, it would be, as root,
insmod /pathtomodule/modulename
That way you won't have to reboot your system.
Once you have wicd installed it should make hooking up to Internet as easy as it gets.