I've used slapt-get before, and it's pretty good. There's another unofficial package manager that works well, but I don't remember what it's called.
Last I knew, slack* use sysvinit. There's no / e t c /init.d, but scripts are in / e t c /rc-something. It's a slightly simpler setup, if I remember correctly.
"No tilda" is just the beginning. You'll run into lots of other things you want that aren't in the repo. You might find them in outside repos, and I think they've improved the scripts to create your own packages, so it's easier than it used to be, but don't quote me on that. Either way, you're not going to get it with a single, simple command like you can with debian. (I guess in gentoo it's also just a single, simple command, but it takes longer to finish.)
If you want to try something really different, there's this:
http://kolibrios.org/en/Pretty sure you'll only get the basics in that - file manager, text editor and a few other things, but it's pretty cool. Dust off your old 386 and fire it up.