This is the second day of my over the phone English class.

And the tToday's topic is about aour favorite singer.

If you want to use 'and,' you can combine it with the previous sentence:

This is the second day of my over the phone English class, and today's topic is about our favorite singer.

Actually, I don't like songs with lyrics, so I don't know themany singers either.

Instead, I like songs with no lyrics songs, like classics, movie OSTs, animation OSTs, and even game OSTs.

He is one of the world-most famous composers in the movie world, and he makes scores of many movies such as The Rock, Tarzan and Interstella.

Or: He is a world-famous movie composer, and he makes scores of many movies such as The Rock, Tarzan and Interstella.

Recently, he was participated in making scores for a movie called Dune.

I will buy his new album ofrom the movie.

This ioday was the second day of the phone English class.

Changing to past tense because you are writing about a class that has already happened.

And the today's topic iwas about aour favorite singers.

Actually, I don't like songs with lyrics, so I don't know the singers eithnames of singers.

You don't need "actually" or "either" in this sentence. These words are often used to provide contrast to a previous statement.
For example, if a classmate says "I like the lyrics of that new BTS song, what lyrics do you like?", you could reply with, "Actually, I don't like songs with lyrics."

Instead, I like no lyrics songs like classicinstrumental music, like classical compositions, movie OSTs, animation OSTs, even game OSTs.

I'm guessing you mean "classical" music like Beethoven, Vivaldi, etc.?
Also, while there is nothing wrong with using the term, "OST" is not commonly used in English. We generally just say "official soundtrack," or for more instrumental music "a film/movie score."

He is one of thea world-famous composer in the moviefilm world, and he makes the scores of many movies such as The Rock, Tarzan and Interstella.

Recently, he was participated in making the scores for a movie called "Dune."

Whilen watching the movie, I was blown away by his songs.

I will buy his new album of the movieam planning to buy the movie's new album.

Just re-worded to sound a little more natural. Nothing wrong with this sentence as it is! :)

This is the second day of my over the phone English classes.

And the tToday's topic is about aour favourite singer.

Don't start sentences with 'And'.
You can combine sentences though - Today is the second day of my over the phone English classes, and the topic is on our favourite singer.

Favourite = UK
Favorite = USA
.

AI actually, I don't like songs with lyrics, so I don't know theoo many singers either.

Instead, I like nosongs without lyrics songs, like classics, movie OSTal music, movie soundtracks, animation OSTsoundtracks, even game OSTsoundtracks.

Soundtracks sounds a bit nicer.
- like classical music, movie, animation and game soundtracks.

He is one of the world-famousmost renowned composers in the movie world, and he makescomposes the scores of many different movies such as The Rock, Tarzan and Interstellar.

Recently, he was participated in makingwriting the scores for a movie called Dune.

I heard they got the sound of the worms by swallowing a microphone and recording the inside of the persons mouth!

When watching the movie, I was blown away by his songsmusic.

For scores, you would call it music as opposed to songs. Unless the music was in 3 minute chunks and arranged like a typical pop song.

I will buy his new album of the movie.'m planning to buy his new soundtrack album (when it comes out.)

It's typically English to talk about your 'plans' when speaking in the future tense.