I enjoy watching TV shows about drama or documentaries about science.

more normal would be "TV dramas or documentaries about science."

This show isn't only about gossip, it also permits me to listen to different accents of English.

or "different English accents"

There are people from different countries: the U.S., Venezuela, Panama, Mexico, Russia, Bulgaria, Africa, et cetera.

I started watching it in Spanish; after, then in English with subtitles, and today, finally, I could understand the majority parof it without subtitles!

not incorrect originally, but this sounds more natural

I really wanted to make the exercise more productive tofor my knowledgelearning, so I wrote on my notebook a summary about each scene in my notebook.

the prepositional phrase usually goes at the end of the thought:

For example:

I gave to my friend a gift.

would be better as:

I gave a gift to my friend.

or

I sang in my head a song.

would be better as

I sang a song in my head.

I'll try again the next time.

We would say "the next time" to introduce a specific thing, like "the next time I have extra free time" or "the next time it comes on TV"

if there is not an unspecified instance, then just "next time" sounds more natural.

I was practicing my English listening (ability) today.

Optional to include ability, but I think it sounds a little better with it included.

I enjoy watching TV shows about dramas or documentaries about science.

Today I watched an episode of the 90 Day Fiancé.

This show isn't onlyjust about gossip, it also permits me to listen to different accents of English.

There are people from different countries: the U.S., Venezuela, Panama, Mexico, Russia, Bulgaria, Africa, etcetera.

In very formal contexts (like a scientific paper), you might see "etc." expanded out to "et cetera", but it's unusual in more standard writing to not abbreviate it. When it is expanded out, there is a space between the two words.

I started watching it in Spanish; after that in English with subtitles, and today, finally, I could understand the majority part without subtitles!

"majority" is a noun on its own, you don't need "part".

The advantage is that they show short scenes about different situations, so it makes it easier to catch the topic.

I really wanted to make the exercise more productive tofor my knowledge, so I wrote on my notebook a summary about each scene on my notebook.

I'll try again the next time.

I enjoy watching drama TV shows about drama or documentaries about science.

Drama is actually a TV show genre so you can use it as an adjective here :) You could also say “science documentaries”.

Today I watched an episode of tThe 90 Day Fiancé.

Since “the” is the first word in the title, it must be capitalized

This show isn't only about gossip, it permitallows me to listen to different accents of English.

What you wrote was perfectly fine, but the word “permit” is used for more formal/serious contexts, like permitting someone to access something classified, permitting a student to use the bathroom, etc.

There are people from different countries: like the U.S., Venezuela, Panama, Mexico, Russia, Bulgaria, Africa, etcetera.

The colon is not needed
Additionally, when we use the word “etcetera” in English we almost never write out the full word. It’s just a thing you can do if you want to sound more fluent

I started watching it in Spanish; after, then in English with subtitles, and today, finally, I could understand thea majority parof it without subtitles!

1. The semi-colon is used incorrectly here and makes the sentence sound a bit cluttered and confusing
2. “The majority part” isn’t grammatically incorrect. You should say “a majority of ___” in this context

The advantage ismost helpful thing is that they show short scenes about different situations, so it makes it easier to catchunderstand the topic.

1. The word “advantage” sounds odd here
2. What you wrote at the end was fine, but “to understand” sounds better

I really wanted to make the exercise more productive to my knowledge, so I wrote oin my notebook a summary aboutof each scene.

1. “To my knowledge” sounds wrong, and the sentence is perfectly fine without it, so I just took it out
2. When talking about writing on something like a book, we use the word “in” instead of “on”
3. “About” works fine, but “of” sounds better

I'll try again the next time.

I’m not sure what you mean here, as the phrase “I’ll try again” is normally used after failure