Today, I woke up at 6:00 a.m. and began my day by taking a bath and having breakfast.

By 8:45 a.m.,

2 hours later, I began my study session after 2 hours , I wrote , which included writing in my daily journal .

"2 hours later" is clarifying information, so you can add it as I have, enclosed with commas and it needs to be next to the "by 8:45," either before or after. Otherwise, in the way you have it, it sounds like an additional two hours have passed, or that your study session was two hours

-->By 8:45 a.m., 2 hours later, I began my study session, which included writing in my daily journal.
OR
-->Two hours later, by 8:45 a.m., I began my study session, which included writing in my daily journal.

After that, I learned five new English words and used them in sentences .

At 11:30 a.m.,

I made my lunch and haved it .

Then I took a little rest , at 3:00before I began my another study session .at 3:00 p.m.. After completeding my study session , I went to the playground to play cricket . After that I returned to my room.

Today, I woke up at 6:00 a.m. and began my day by taking a bath and having breakfast.

By 8:45 a.m.

this doesnt not make sense as a standalone sentence

I began my study session and after 2 hours , I wrote in my daily journal .

Doesn't make sense, 2 statements without something to connect them

After that, I learned five new English words and used them in sentences .

Tense is super important, please keep that in mind. Also DO NOT learn 5 words a day, learn like atleast 20. If you do that for 1-3 years, your English is going to be WAY better.

I made my lunch and havate it .

Then I tooknapped for a lbittle rest , a. At 3:00 I began my another study session .After I completed my study session , I went to the playground to play cricket .After that I returned to my room.

fellow south asian? yoo, im malayali

Day- 5

Today I woke up at 6:00 a.m. and began my day by taking a bath and having breakfast.

Another option: Today I woke up at 6 am and started my day with a bath and then breakfast.

By 8:45 a.m.,

Also fine: "By 8:45 am,"

I began my 2-hour study session after 2 hours , I wrote , which included writing in my daily journal .

The timing isn't clear here. If this 2 hours is referencing the time you spent studying (including journaling), the above works

After that, I learned five new English words and used them in sentences .

At 11:30 a.m.,

I made my lunch and have it ate.

I'd normally just say "I had my lunch." You only need to say that you "made" lunch if you want the reader specifically to know that you prepared the lunch that you had.

Then I took a little rest , a. At 3:00 p.m., I began my another study session . After completeding my study session , I went to the playgroundark to play cricket . After that, I returned to my rohome.

Since you used "a.m." each time above, it feels strange not to have "p.m." here.
The "playground" is an area in a park that has swings, slides, etc., for children to play on. But you would play cricket in a field. Both are located at the park, so you reference the park here.
It's typically not necessary to specify what room you went back to, we would normally just say that you "went back home"