First, I'm not so sure that Slackware could ever be seen as a newbie distro, however, there's nothing *too* wrong with that and now you've got it, there's no reason NOT to try it.
Secondly, just boot from Slax and see how far you get. 99% of ethernet cards "just work" in Slax/Slackware (as for any other linux system) but they may need some configuration if you do not have a DHCP system.
DHCP give out an address to any ethernet card on your local network that asks for it. If you have a DHCP server somewhere (usually on things like broadband routers etc.) that Slax will just pick up on that and use it and hey presto! Internet access.
If you don't have DHCP, the above just won't work (you'll run a browser on Slax and get nothing but errors) so you have to set the card IP addresses manually. If you've not done it before (in Windows or otherwise), I'd suggest learning how to do that before you try any sort of Linux as they will all need those settings to be made.