Learn Polish: Oglądam nowy film z córką. - I am watching a new film with my daughter.

Because of the word z (with). When you do something "together with" someone, the preposition z forces the following noun into the Instrumental case. For feminine nouns that normally end in -a, that ending changes to -ą.

The English translation says "with my daughter". Why isn't the Polish word for "my" in the sentence?

In Polish, if the ownership is obvious from the context, you usually drop the possessive word. Since the speaker says oglądam (I am watching), it is automatically assumed they mean their own daughter unless they specify otherwise.

Since nowy film is the thing being watched, shouldn't its ending change?

It is indeed the direct object, which means it is in the Accusative case. However, for masculine non-living nouns like film, the Accusative form is identical to the basic dictionary form. The adjective nowy also stays the same to match it!

Is there a rule for when to use z versus ze for "with"?

Yes! You generally use z, just like in z córką. The form ze is only used to make pronunciation easier when the next word starts with certain awkward combinations of consonants, such as ze mną (with me).