In my view, both Googlegroups and Yahoogroups are almost as good. I used to favour Yahoogroups first, then took to Google and have now been involved with the launch of http://groups.yahoo.com/group/fsug-goa Mailman is the right choice, it’s Free Software. But you need to be able to administer it and have a fast-enough server.
Of the earlier two, each has one or two advantages. Yahoo has a add-10 addresses a day limit. But Googlegroups will block you if you add too many. Don’t worry, not much differences between the two.
Facebook (which is getting a lot of groups-like features these days) is good too. But, at best, it can supplement a Yahoo/Googlegroup. I don’t think Facebook is good enough as a stand-alone option, because it is easy to ignore (the discussions are less interactive, people can just be on the ‘group’ but not active).
In a word, go for either Googlegroup/Yahoogroup and take Facebook to supplement. Or Mailman if you can.