"First entry

Well this is my very first diary (diary: 日記; is that what you meant?). I'm seventeen years old from Japan. I'm really willing to improve my English more and more!
It rained quite a lot today, so I didn't feel good today. I hope it will be sunny tomorrow."

-> You don't need to repeat the "today"! You can either say:

-It rained quite a lot today, so I didn't feel good.

or

-It rained quite a lot, so I didn't feel good today.

We have similar names! How fun! One of my best friends is called Marina too! (•͈ᴗ•͈✿)

First entry

Well this is my very first diary. I'm seventeen years old from Japan. I'm really willing to improve my English more and more!
It rained quite a lot today, so I didn't feel good today. Hope it will be sunny tomorrow.

Overall, this is nice! But don't repeat "today" if we know the main subject!

First Entry

Well this is my very first diary. I'm seventeen years old, and I'm from Japan. I'm really willing to improve my English more.
It rained quite a lot today, so I didn't feel good. Hope it will be sunny tomorrow.

Overall, pretty good! Keep up the good work :D