Next Tuesday is the biggest holiday in Korea thatwhich is the first day of January in Tthe lunar calendar.

Because of the pandemic, the government doesn’t recommend to gogoing to hometowns.

However, my kids really want to meet their grand parents, so we finally coame here.

My kids also enjoy withe dogs and cows in my hometown.

Next Tuesday is the biggest holiday in Korea that is, the first day of January in Tthe lunar calendar, is the biggest holiday in Korea.

Because of the pandemic, the government doesn’t recommend to go toing back to your hometowns.

However, my kids really wanted to meet their grand parents, so we finally come herewent back.

We are getting together and talking about our lives.

"We are" is often said/written as "we're".

My kids also enjoy withe dogs and cows in my hometown.

We will stay three days here, and I hope my kids can havmake good memories.

"We will" is often said/written as "we'll".

Next Tuesday is the biggest holiday in Korea that is: the first day of January ion Tthe lunar calendar.

Because of the pandemic, the government doesn’t recommend to go tohat people go to their hometowns.

The most common way to express this idea, at least in American English, is to say "the government advises against traveling to see family over the holidays."
Sticking closer to your original sentence, one always uses a possessive adjective when referring to a hometown: my hometown, their hometown(s), etc.

However, my kids really want to meet their grand parents, so we finally come here.

My kids also enjoy wilike the dogs and cows in my hometown.

Other options: my kids also have fun with the dogs, have a good time with the dogs, enjoy playing with the dogs

We will stay three days here, and I hope my kids can havwill have some good memories of their trip.

I'm not sure how to explain why you need to use "will" instead of "can" here, but "can" sounds odd to me as a native speaker.
It's good to add "some" before "memories" to clarify that you mean "memories" in the sense of "recollections" rather than "the capacity to remember." Adding "of their trip" sounds more natural and also helps clarify that these are recollections rather than the capacity to remember.

Next Tuesday is the biggest holiday in Korea that is the first day of January in The lunar calend: on the Lunar calendar, it's the first day of the year.

We generally say "on the calendar," not "in the calendar."

Because ofDue to the pandemic, the government doesn’t recommend to go to hometownsreturning to our hometowns for the holiday.

However, my kids really want to meet their grand parents, so we are finally come heregoing back.

We are gettingplanning to get together and talking about our lives.

You could also say "We will get together and talk about our lives," because you're talking about something will happen in the near future.

My kids also enjoy wiseeing the dogs and cows in my hometown.

You could use "visiting with" or "playing with" here as well.

We will stay here for three days here, and I hope my kids can havwill make good memories.

Next Tuesday is the biggest holiday in Korea, and that is the first day of January in The lunar calendar.

Because of the pandemic, the government doesn’t recommend to go togoing to your hometowns.

However, my kids really wanted to meet their grand parents, so we've finally come here.

My kids also enjoy wiseeing the dogs and cows in my hometown.

We will stay here for three days here, and I hope my kids can havmake some good memories.

Next Tuesday is the biggest holiday in Korea that is, the first day of January in Tthe lunar calendar.

Because of the pandemic, the government doesn’t recommend to gogoing to hometowns.

However, my kids really wanted to meet their grand parents, so we finally cocame here.

We arehave been getting together and talking about our lives.

My kids also enjoy playing with dogs and cows in my hometown.

We will stay here for three days here, and I hope my kids can havmake good memories.