Click the Add button above your contact list, or Contacts > Add a Contact, and type your friend's Yahoo! ID. They get a request, and once they accept, you see each other online.
Click your status under your name to set a status message. Your friends see it in their contact list and in their Y! Updates tab.
Messenger > My Display Image lets you share a picture or a drawn avatar.
Go invisible from your status, appear offline to one friend with Stealth Settings, or ignore someone who bothers you. See also reporting abuse.