First dialy~ry entry

'diary' is the entire journal, composed of many entries.

This was my first dialry entry in English, so I was little nervous...(C (can it be true that I should write sentences or comments in past tense when I write dialry entries in English?).

To answer your question: depends if you're describing the past or not. Fine to use any tense if it is appropriate.

I didn't know what I have to write first, so I introduced myself!

I liked anime and manga.

Past tense here implies you no longer like anime and manga. If you do still, use present tense.

Especially, "MY HERO ACADEMIA", which was one of my favorite arts...!mangas.

Can't use the word 'art' in plural to refer to individual works.

I studyied English to be able to express my opinion in English and to be able to read arts in English.

Removed redundancies.

First dialRy~

This was my first dialRy in English,so I was little nervous...(CanIs it be true that I should write sentence or comments in past tense when I write dialry in English?)

No, you don't need to write in past tense! Feel comfortable to write in present tense, unless you want to actively practice past tense
Write in the tense (present/past/future) you would write in your native language!

I studyied English to be able to express my opinion in English and to be able to read arts in English.

First dialry~

英語ではLとRが違う文字なので気をつけてください！
"Diary" is only ever spelt with an R.

This was my first dialry in English, so I was'm a little nervous...(Can (Is it be true that I should write sentences or comments in past tense when I write dialyry entries in English?)

You should use past tense if you're describing something that happened in the path. Here you're writing about something that you (at the time of writing) feel in the present, so you should use the present tense here.
過去にあった事を書いているなら過去形を使うべきなんですが、ここは書く時の感情を表しているので現在形が最もふさわしいと思います！

I didon't know what I have toshould write first, so I'll introduced myself!

"Have to" implies that you are obliged to do something, while "should" feels better here because it implies a choice
「have to」は「しなければならない事がある」みたいに聞こえますよ。

I liked anime and manga.

Here saying that you "liked" it gives the impression that you no longer like it. You should generally write statements like this in present tense.
「liked」は過去形なので「以前は好きだったけど…」みたいな印象をしますよ。こういう文は現在形を使いましょう。

Especially, "MY HERO ACADEMIA"wa which is one of my favorite artspieces of media...!

Like the previous sentence, using past tense here gives an impression that it is not one of your favourites now.
以前の文みたいに、過去形の「was」を使うと「好きだったけど…」の印象をします。
"arts" is a special term that describes the different methods of creating art. When talking about a singular thing (作品) "piece of art", "work", or "piece of media" is probably the most appropriate way of expressing it. Here I would use piece of media, but any of these could work!

By the way, since I'm British I normally spell "favorite" as "favourite". This isn't something I will correct though.
ちなみに、イギリス人なので普通は「favorite」を「favourite」でスペルします。添削すべきな事じゃないけど…笑

I am studyeding English to be able to express my opinions in English and to be able to read artsmedia in English.

Since this is something that you are currently doing, you should use present continuous tense.
今している事なので現在進行形を使いましょう。
"Opinion" is singular, but you probably have a lot of opinions that you'd like to express so plural is probably the best option here!
「Opinion」は単数形なんですが、多分たくさんの思いがあるので複数形が最もいいと思います！

I did'll do my best!

First dialy~

First diary entry~

This wais my first dialry entry in English, so I was'm a little nervous...(Can it be (is it true that I should write sentences or comments in past tense when I write dialymy diary entries in English?)

For events that already happened in the past, you can use the past tense. When you say “This was my first diary entry in English” it sounds like you might be showing a diary entry you wrote some time ago. So in here, use the present tense instead.

I didon't know what I have to write first, so I'll introduced myself!

I liked anime and manga.

「liked」を使うと、今はアニメと漫画がもう好きじゃないみたいです。

Especially, "MY HERO ACADEMIA"wa is one of my favorite arts...!series.

"arts" isn't really used to refer to anime. You can just use "anime" or "series".

I studyied English to be able to express my opinion in English and to be able to read arts and read comics in English.

「arts」って、英語のコミックのことですか？もしそうなら、「comics」で大丈夫ですよ。

You can use past tense here if you mean to say that you studied English in the past and that you're not studying anymore.
If you're still studying English, you may say "I'm studying English so I can express my opinions and read comics in English."

I did'm doing my best!

「did」を使うと、もう英語の勉強は終わったみたいです。 もし今も英語を勉強しているなら、「I'm doing my best!」を使ってください。