Hi, I'm not a native speaker, but as far as I know, you would use être with verbs that indicate a change of state or movement. There is an acronym to help you remember them, "DR & MRS VAN DER TRAMP" (devenir, revenir, monter, rester, sortir, venir, aller, naître, descendre, entrer, retourner, tomber, rentrer, arriver, mourir, partir)

I think French native speakers will answer this question better than me, but as far as I see, etre is used with exception verbs like entrer, sortir, arriver, partir, aller, venir, descendre, naitre, mourir, monter, tomber, rester, with all their derrivatives (i.e. revenir, devenir), with all reflexive and passive verbs. Avoir fits with everything else.

I remember from my school days (it's 20 years ago so don't quote me) that there are something like 16 verbs which take être. You can surely find a list of them somewhere online.