The summer vacation has started!
Today, summer vacation began!
Now, I am going to my parents house. They live in Deagu. 
I go there by train named Mugunghwua. It takes about 4hours from Seoul to Deagu.
I read book, watch movie in train.

I have a plan to meet with my close friends I haven't seen for a while, and go camping with my family.

I really looking forward this vacation.

The&nbsp;Summer vacation has started! &nbsp;(Just use "vacation" or "my vacation".)
Today, my summer vacation began! Now, I am going to my parents' house. &nbsp;(Note the apostrophe after "parents" to denote possession, with "parents' house". &nbsp;You can also say "I'm on my way to my parents' house.)
They live in Deagu. I am going there by a train named Mugunghwua. (Use the present continuous, since you're in process of going or traveling. &nbsp;I'd use "...called Mugunghwua.")
It takes about 4 hours from Seoul to Deagu. I read a book, and watch a movie on the train. &nbsp;(Note "on the train" which doesn't actually mean "on top of the train" - it's odd that you don't say "in the train", I know.)&nbsp; I have a plans to meet with my close friends I haven't seen for a while, and to go camping with my family. &nbsp;(Note "to have plans to..." &nbsp;Good use of the present perfect! &nbsp;Since you used "to meet", use "to go" in the next clause.)
Well done, practicing those prepositions of place! I really looking forward this vacation.

The summer vacation has started!
Today, summer vacation began! Now, I am going to my parents' house. They live in Deagu. I'm going there by a train named Mugunghwua. It takes about four hours from Seoul to Deagu. I read books and watch movies in train. I'm planning to visit my close friends I haven't seen for a while, and go camping with my family. I'm really looking forward to this vacation.