The Summer vacation has started!  (Just use "vacation" or "my vacation".)

Today, my summer vacation began!
Now, I am going to my parents' house.  (Note the apostrophe after "parents" to denote possession, with "parents' house".  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.  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.  (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.) 

I have a plans to meet with my close friends I haven't seen for a while, and to go camping with my family.  (Note "to have plans to..."  Good use of the present perfect!  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.