Learn Finnish: Kyllä, minä olen opiskelija.

Finnish does not have articles like "a", "an", or "the". The word opiskelija translates to both "student" and "a student" depending on what sounds natural in English.

Is it mandatory to use "minä" here, or could I just say "olen opiskelija"?

You can completely drop minä and just say olen opiskelija. The verb olen already ends in -n, which shows it belongs to "I". Including minä just adds a bit of emphasis or makes it sound a little more formal.

I usually hear Finns say "joo" for yes. Why does this sentence use "kyllä"?

Kyllä is the standard, proper dictionary word for "yes" and is very common in writing or polite speech. Joo is the casual, spoken version of "yes" that you will hear constantly in everyday conversations.

Which verb does the word "olen" come from?

It comes from the verb olla, which means "to be". Finnish verbs change their endings to match the person doing the action, so the base verb olla turns into olen to specifically mean "I am".