I headed to the gym to work out my chest and abs today, after I walked my dog in the morning.

Remember the past tense.

Repeating today again after the last sentence isn't need

I wanted to use the chest adduction machine that can build up the middle part of my chest muscles, but unfortunately there were too many gentlemenpeople waiting for the machine, so I left.

I actually didn't know what the machine was called, I had to google adduction to see what it meant myself :)

After the exercise, I went to the supermarket just next to the gym that I would walk by on the way back to home, and I bought a bunch of fruits like blueberries, strawberries, and even grape tomatoes, since I started my healthy diet to make my tummystomach smaller.

"tummy" is kind of a childish/cutesy phrasing and doesn't fit with the tone of the rest of the piece.

There was a kid who used to do very well with his dad, but he behaved badly today with a woman.

Honestly, I am very bad at remembering their names.

Embarrassing mMoment tToday

Title capitalization rules

Today iwas a usual and ordinary day.

This is now past tense as you are recounting a story from your day => was

A bit more natural: Today was a typical, ordinary day.

I headed to the gym to work out my chest and abs today after I walked my dog in the morning.

No need to repeat "today" as it's understood from the previous sentence.

A bit more natural: I headed to the gym to work out my chest and abs after walking my dog in the morning.

I wanted to use the chest adduction machine that can build upworks the middle part of my chest muscle, but unfortunately there were too many gentlemen waiting for the machine, so I left.

"Gentleman" is being extremely polite (which is fine), but if you wanted to be more casual, you could say "other guys."

After the exerciseing, I went to the supermarket just next to the gym that I would walk bypass on the way back to home, and I bought a bunch of fruits, like blueberries, strawberries, and even grape tomatoes, since I started myas part of my new healthy diet to make my tummy smaller.

"The exercise" makes it sound like you did one form of exercise only.
"Tummy" is a cute word that kids and women would use, but men would typically avoid (in case you care about that; I don't, personally).

I cookedmade spicy instant noodles and fried eggs after I wenwhen I got back home.

More natural

I did not want to cook the instant noodles, because they make me fat.

I was just too lazy to make other food for lunch.

Better: I was just too lazy to make something else for lunch.

Then I went tout for work after taking a nap.

I am a coach, and I haved three lessons today.

We are still in the past tense for your story

The second class is/was for (tiny tossers?), so the parents would stay/stayed with their kids during the class.

This can remain in present tense (is, stay) because it is a current description of the class. However, I would still opt for "was, stayed" as continuation of the story

I don't know what "tiny tossers" refers to, this calls for a brief description e.g, "... for tiny tossers, which describes XXX,"

There was a kid who used to do very well with his dad, but he behaved badly today with a woman.

Then, she replied to me that she is his mom.

More natural: Then she replied that she was his mom.

What an embarrassing moment I had today.

Alternative: That was my embarrassing moment from today.

I am so sorry tofeel bad about mixing the kids up.

Using "I am so sorry" needs to be directed at someone, and here there is no one to direct it to except for us, the readers, which doesn't make sense.

Honestly, I am very bad at remembering their names.

Today is a usualwas and ordinary day.

"usual and ordinary" is an awkward phrasing. Also, this seems to be in the past, so "was" is more appropriate

I wanted to use the chest adduction machine that can build up the middle part of my chest muscle, but unfortunately there were too many gentlemenpeople waiting for the machine, so I left.

After the exercise, I went to the supermarket just next to the gym that I would walk by on the way back to home, and. I bought a bunch of fruits like blueberries, strawberries, and even grape tomatoes, since I started my healthy diet to make my tummy smaller.

There was a kid who used to do very well with his dad, but he behaved bad today with a woman.today he was with a woman and behaved badly.

"behaved bad with a woman" implies that his behavior was towards the woman, rather than the woman accompanied him.

EAn embarrassing moment today

I headed to the gym to work out my chest and abs today after I walked my dog in the morning.

After the exercise, I went to the supermarket just next to the gym that I would walk by on the way back to home, and I bought a bunch of fruits like blueberries, strawberries, and even grape tomatoes, since I started my healthy diet to make my tummywaist smaller.

“tummy” sounds very cute. Something like “waist” is more fitting with the rest of your writing.

I did not want to cook the instant noodles, because they will make me fat.

Then I went tout for work after taking a nap.

I am a coach, and I havetaught three lessons today.

“have” sounds like you are taking the lessons. “taught”, “instructed”, or “coached” are better choices :)

The second class is for tiny tossddlers, so the parents would stay with their kids during the class.

“their” is a bit more natural.

There was a kid who used to do very well with his dad, but today he behaved bad todaly with a woman.

Honestly, I am very bad at remembering their names.