Learn Russian: Я люблю слушать нашего нового преподавателя.

In Russian, the verb слушать (to listen) takes a direct object in the Accusative case, without any preposition. You don't need a word for "to"; you just listen directly to the person or thing.

Why do the words нашего and нового end in -его and -ого?

This is the core rule for masculine animate modifiers in the Accusative case. Because the teacher is a living person (animate), adjectives and pronouns describing him take the special endings -его and -ого. If the object were an inanimate thing, like a podcast, they wouldn't change from their dictionary forms.

How should I pronounce the letter "г" in нашего and нового?

Whenever you see the endings -ого and -его on adjectives and pronouns, the letter г is pronounced like an English "v". They are pronounced roughly like "na-she-vo" and "no-va-va".

The dictionary word for teacher is преподаватель. Why does it end in -я here?

Because it is an animate masculine noun in the Accusative case. For masculine nouns that end in a soft sign (-ь), the soft sign changes to -я when the noun becomes the direct object.

Could I use учителя instead of преподавателя?