Hi 7Cookies,
error better [optional] notes

I am so excited today.

Today is really a exciting day!
I had my winter vacation on Jan. 13, but I needed to stay at school and take some extra English lessons. Yesterday my lessons were finally over and today, after about 6 hours [on the] train, I am at home today.

because you have already mentioned "today" in the sentence, it would flow better as "I am home at last" instead of re-using the word "today".

Yesterday my lessons were finally over and today, after about 6 hours on the train, I am home at last.
or
Yesterday my lessons were finally over and, after about 6 hours on the train, I am finally home today.

I never have felt so much affection for my parents and my home. It's so warm, so beautiful, so...whatever, I am so happy because I am at home now.

Tomorrow will be the [start of the] Chinese New Year's Eve, which is called chuxi(除夕) in Chinese. Our big family, including my grandparent, my parents, my aunts my uncles and my cousin, will be together. We will have an abundant and delicious family reunion dinner(年夜饭) and we will watch the Spring Festival Party on CCTV-1 together, while waiting for the Chinese New Year/the Spring Festival, and there will be a family party too. I can't wait now!

Notes:
    never start sentences with "and", "but" or "because", it's considered bad form.

Cheers,
Andrew