I usually get up at six am6 AM, and then, boil water, brush my teeth, wash my face, and do za-zen.

Zazen is a noun in English.

It's a specialized term here. Usually, I'd only hear it around Zen meditators (if you're in a temple around other Zen people, you can also say "to sit", but that would be confusing if you use it in general).

"and meditate" would work better for a general audience.

After za-zen, I change clothes, and pour the plain boiled water into a vacuum cup, and then, drink water while attending class.

After the class, I go to the gym and lift a barbell.

"lift weights (as in "I go to the gym and lift weights") is more common. You can also say just "lift" or "lift barbells". People will understand "lift" since you mentioned the gym.

AndThey talk about the workout plans for that day.

Afterwards, I go home and. On the way, I buy bread for my aunt by the way or buy discount vegetables for myself.

For "and", it's nice to have things in a chronological order.

When I return home, I pour a cup of plain boiled water and attend a class until 1 pm.

And tThen, I eat, play with my little niece, and take a break.

For English, you always want to specific the subject ("I" here). A sentence should say who or what is doing something.

"I eat, play with my little niece and take a break" is correct as well. The last comma in a list of three things is the Oxford comma. You don't need it, but it can make things less confusing.

I usually get up at six 6am, and then, boil water, brush my teeth, wash my face, and za-zen.

"six am" is how you would say it, but since "am" is an English word, it's best not to write this. (I've never heard of "za-zen"; maybe "meditate" is a better choice of word.)

After za-zen, I change clothes, and pour the plain boiled water into a vacuum cup, and then, drink waterwhich I drink while attending class.

The word "plain" is redundant. In school we're told to avoid the repetition of the word "then"---generally only children have more than one "then" in a sentence (even though it's grammatically correct). (I don't know what a "vacuum cup" is.)

After the class, I go to the gym and lift a barbells.

Here, even if you lift only one barbell, to refer to the exercise we should say "lift barbells" (as "lift a barbell" sounds like you just lift it once, and that's it).

And they talk about their workout plans for that day.

Generally, we're discouraged from starting sentences with "And". It's not a strict rule, but they usually read better if merged into the previous sentence. In this case "...greet each other every day, and talk about their workout plans..." sounds much smoother.

Afterwards, I goon the way home andI buy bread for my aunt by the way or buy discount vegetables for myself.

"by the way" is not correct choice of term (it's "on the way")---"by the way" indicates a change of topic. E.g. "By the way, did you hear about my birthday party?"

When I return home, I pour arepare another cup of plain boiled water and attend a class until 1 pm.

"pour" refers to the action (i.e. you're telling the reader about your pouring action, not the preparation of your drink); I feel "prepare" is more precise.

AndI then, eat, play with my little niece and take a break.

There's no subject in the original sentence.