How do I know which verbs take être in the passé composé.

Well, the easiest way to remember this is...For almost all non-reflexive verbs, you will use 'avoir' rather than être. There are however, some exceptions. A particularly useful way of remembering which ones take être is the anagram DR & MRS VANDERTRAMP. Each of these letters stands for a different verb, which you can note down and learn off by heart - that way you'll never forget. If the verb, however, is reflexive, then it will always take être. Hope that clears things up. In short, avoir most of the time, but être with the exceptions of Dr and Mrs Vandertramp and reflexive verbs. For example, it would be je suis allé rather than j'ai allé, because aller is the one of the two 'A's in the anagram.