Can you correct my first informal letter?

Dear Mark,
Thanks for your letter. I'm very glad to hear you're well.
Sorry it's taken me so long to write but I was busy preparing for my exams.
Yesterday I saw the last Tarantino's movie. It was amazing. I really liked it from the beginning to the end, and I found all the actors perfects for their roles.
Did  Have you already seen this movie? What you think about it? Let me know.
This week I have started learning Japanese. It's a very interesting language, but it's also very difficult. I learned the Hiragana alphabet and next week I'm going to learn the Katakana alphabet.
As you know, my dream is to live in Japan. I love the Japanese culture, the food and the people who live there.
And you? Are you learning a second language?
Now I have to go back to my homework now.
I'm looking forward to your next letter.

Best wishes,
Giuseppe

Can you correct my first informal letter?

Pretty good, note that proper nouns get capital letters at the start.

The word "last" is associated with the "movie" not with Taratino so put them together other wise you mean there are many Tarantinos rather than many movies made by Tarantino.