It’s a lot easier to get into then it used to be! Most BBS systems support connecting over the internet using the telnet protocol. Some of them you can connect to right off their web page.
You can find everything you need over at the telnet bbs guide:
Grab yourself a copy of SyncTerm for whatever platform/OS you use. It has a built-in list of BBS addresses, start clicking on them, sign up for a few, check out the Telnet BBS Guide on the internet as well. Have fun!