I love going to watch sports games. Especially, I like to watch baseball matches. 

-> I love watching sports games especially baseball matches. (Remove "going to" since you're stating a fact.)

When I was in elementary school, I attended my first baseball game with my father. (Good sentence)

at that time, Hanwha Eagles Baseball Team was based in my hometown. so, naturally I cheered up Hanwha Eagles Baseball Team.

-> That time, Hanwha Eagles Baseball Team was based in my hometown so naturally I cheered for Hanwha Eagles Baseball Team. (Remove AT at the beginning of your sentence as well as period and comma before and after SO.)

Because I was so young, I didn’t the rule of the baseball, but I felt the passion for winning our team.

-> Because I was so young, I didn’t know the rule of the baseball but I felt the passion for winning by our team. (Include KNOW so you can complete the rest of your sentence and remove THE before baseball.)

also It was very fun to be a part of the crowd.

-> It was also very fun to be a part of the crowd. (Take note of your word order and remove A before part.)

When our batter hit an amazing home-run in the 9th inning, I was very exciting!

-> When our batter hit an amazing home-run in the 9th inning, I was very excited! (Use EXCITED rather than EXCITING since you're describing yourself and not the game.)

It was an unforgettable game watching experience. (Good sentence)