How do I remember which verbs take être in the perfect tense?

There are a selection of "verbs of movement" in French which use être as their auxiliary verb in the perfect tense instead of avoir. You can remember these using the mnemonic "MRS VAN DER TRAMP":

Monter - to go up
Retourner - to return
Sortir - to go out

Venir - to come
Aller - to go
Naître - to be born

Descendre - to go down
Entrer - to enter
Rentrer - to return

Tomber - to fall
Rester - to stay
Arriver - to arrive
Mourir - to die
Partir - to leave

These verbs all use être instead of avoir in the perfect tense: e.g. Je suis allé(e), rather than j'ai allé. As well as the verbs of movement, all reflexive verbs also take être: e.g. Il s'est couché. When using être as the auxiliary, the past participle must agree with the subject of the verb: e.g. Nous sommes montés.