Learn Spanish (Latin America): El director dice que me lo va a confirmar hoy. - The director says that he is going to confirm it to me today.

Yes. In English, we often drop the word "that" and just say "The director says he is going to confirm it." But in Spanish, you must always include que to connect the reporting verb (dice) to what was said.

What do me and lo mean here, and why are they together?

me means "to me" (the indirect object) and lo means "it" (the direct object). When you have both, the indirect object (me) always comes before the direct object (lo). Together, they mean "it to me".

Could I put me lo somewhere else in the sentence?

Yes! Because the second part of the sentence uses an infinitive verb (confirmar), you can attach the pronouns directly to the end of it. You could say va a confirmármelo. Notice that when you add them to the end, an accent mark is added to keep the pronunciation stress on the right syllable. Putting them before the conjugated verb (me lo va a confirmar) is also perfectly correct and very common.

If the director is talking about the future, why do we use dice (present tense)?

Because the action of him telling you this is happening right now. We use dice (he says) to introduce the statement, and then va a confirmar (he is going to confirm) to show exactly what he promised for the future. Since the reporting verb is in the present, the tense of the reported quote doesn't need to change.