Learn Spanish (Latin America): ¿Te importan los subtítulos?

This is because importar works just like gustar. In Spanish, you aren't actually saying 'Do you care about the subtitles?'. Literally, you are asking 'Do the subtitles matter to you?'. Since los subtítulos (the subtitles) is plural, the verb must be the plural importan.

What does te do in this sentence?

Te means 'to you'. It tells us who the subtitles matter to. With verbs like importar and gustar, we use indirect object pronouns like me, te, le, nos, and les instead of subject pronouns like tú or yo.

Could I put the subject first and say '¿Los subtítulos te importan?'

Yes, you can! However, in Spanish, especially in questions and with verbs like gustar or importar, it is much more natural and common to put the verb before the subject.

How would I say 'Do you care about the movie?' instead?

You would say '¿Te importa la película?'. Because la película (the movie) is singular, you must change the verb to the singular form, importa.