Slackware uses it's own packaging system. Not too much different than standard tarballs with some extra "zing"
, but it's own nonetheless. You can look into options such as autopkg (which I could not find a decent link for), but for the most part I suppose you won't find the same ability and ease as you have with apt-get. Personally I don't like apt-get. Not because I'm a masochist, nor do I think things shouldn't evolve, but apt-get doesn't give you the power that a standard tarball does (not in the same way anyway). You don't have the ./configure options, you cannot edit the Makefile to taste, and more importantly, you lose the ability of knowing every single package that is on your system since you aren't installing it. Eventually you come across packages you think you don't need, remove it and screw up something major. In the end, you didn't save time when you end up troubleshooting the hell out of everything.
I really like the slackware packaging system, and as long as you can view it as it's own thing, and leave apt-get back where it was you will be fine. Once you install slackware, fire up pkgtool and you will see how easy slackpacks are to install.