First notebook entry: Introducing myself

I speak Japanese and am learning English and Spanish.

or

I speak Japanese and I'm learning English and Spanish.

In the future, I wanna speak French too.

During my holiday (Please make corections)

During my holiday, I didn't really go to so very many places, because I had to move into a my new place. *"a" is correct, but i think "my" sounds better because it emphasizes you're moving into a new living space.
So the first week of the summer vacation, i did lot's of packing and cleaning, and the second week I spent unpacking and cleaning.
During that time, I was excited and at the same time exhausted.
I was excited because we were moving to a new place and i was exhausted because of the cleaning and from carrying lot's of boxes in and out of the house.
My summer vacation started weird but it was not so bad!!
Well... the rest of my summer vacation was relaxing and fun so... I guess it turned out to be okay.
I was very excited (most excited??) when my grandparents came to visit in early August.
They stayed here for 2 weeks. We took them to nearby tourist spots.
They really enjoyed the castle.
The Zoo next it was very nice too. It was very different from the Zoos i saw before, the animals there looked very healthy and relaxed.
I really liked that particularly Zoo.
( I usually hate Zoos cuz animals gets locked up in a tiny cages)
I had lot's of fun there.
I also visited my friends house, who i know from my previous school.
WE went shopping together in the city, and we also did artistic facial make-up. I had lot's of fun with my friend as well.
Time passed by so fast and the summer vacation was over in a matter of a second.
I wished that it would have lasted longer.

You also want to leave out the apostrophes in "lots". 

 Your English is so good!