If "avere" is used with transitive verbs, which take a direct object, then in the instance above, what is the direct object? What am I missing? Or is this some kind of exception? I notice that essere is used in "La festa è iniziata ieri." I can see that this latter case is intransitive and why, therefore, "essere" is used. Does it have to do with the passive voice?
When a party has started, it means that the party itself has begun. The party has not started some kind of action like a sentient being; it's intransitive. That's easy enough, but when we see "ho iniziato bene", it's not clear what is being started. However, since "I" is the subject, we know that I have started some kind of action or activity. We don't know what that action/activity is, but we know that it exists, and that's what tells us that the verb is being use transitively.
Yes, I think so. If it's a person starting the action, then the verb is being used transitively (they are starting something, so there is a direct object even if it's not mentioned.)
ho cominciato = I started (something) (transitive)
è cominciato = (something) has started (intransitive)