The difference between groups and pages is a little confusing. I joined the following groups: "FaceBookAppsForLibraries", "Stereogum" and "I Remember When the TC City Hall Was A Food Lion!". I think that last one is a page instead of a group, because you join as a fan rather than a member.
Oh, and I also set up an event for Father's Day and sent it to my family members on Facebook. That looks like a good feature.