Minä ostan lipun elokuvaan.

In Finnish, verbs are conjugated for each person, so the subject is normally clear from the verb ending. Minä means “I” and is optional. You include it for emphasis or clarity. In everyday speech you’d usually say simply:
Ostan lipun elokuvaan.

Ostan is the first-person singular present tense of the verb ostaa (“to buy”).
• ostaa = infinitive (“to buy”)
• drop -a/a from the infinitive → ost-
• add the -an ending for “I” → ostan (“I buy” / “I am buying”)

Lippu is the nominative form (“a ticket” as a subject). Here it’s a specific, complete object of the verb, so Finnish marks it with -n (the so-called accusative/genitive object form).
• nominative (subject): lippu
• accusative/genitive (singular, complete object): lipun

Lippua is the partitive singular. You use the partitive to express:
• an incomplete action (e.g. “I’m buying some ticket(s)”)
• an unspecified quantity (“some ticket”)
• ongoing, habitual actions
If you want to say “I’m buying a ticket” as a complete transaction, you use lipun. If you just mean “I’m buying some tickets” in general, you’d say ostan lippuja (partitive plural).

Yes, you can form a compound noun elokuvalippu (“movie ticket”) and say Minä ostan elokuvalipun. In that case:
• elokuvalippu (compound) becomes elokuvalipun in the accusative/genitive (complete object)
Both sentences are correct; the original stresses the destination (elokuvaan), this one focuses on the type of ticket.

In Finnish you invert the verb and subject and add -ko/-kö to the verb. Often you drop Minä.
Ostan lipun elokuvaan → Ostanko lipun elokuvaan?

You’d pluralize both lippu and elokuva. Since you’re specifying multiple complete tickets and multiple movies, use the accusative/genitive plural -t on lippu and the illative plural -in on elokuva:
Ostan liput elokuviin.