Yesterday was themy mom's birthday of my mom.

While "the birthday of my mom" is understandable, English speakers tend to put the possessive form before the object being possessed. E.g. "My mom's birthday." "My brother's toy." "John's boyfriend." "Her stethoscope."

I baked a loaf of chocolate chip banana bread for her.

Many objects in the English language are treated as plural, even when they might be a singular object, and as such need a collective noun. In this case bread is treated as plural so we need a collective noun to tell us how much bread: "A loaf of bread" or "A slice of bread." Another example of a plural noun is trousers or pants, you say "I bought a pair of pants".

She liked the banana bread, but she especially loved that I didmade it instead of buying one.

In English we use the verb "to do" for processes which do not create new things, such as "to do the laundry", "to do chores." If you are creating something new then you would say "I made some bread" or "I made a work of art." For bread specifically we use the same verb that was used to create the bread "I baked the bread" would be best.

SMy mother ate a traditional dish of my country.

Because the previous sentence changed the main topic from "my mom" to "my parents" you need to change it back to your mother to prevent the reader getting confused.

As a more general note, this sentence is very formal when compared to your use of "my mom" earlier. "My mom" is familiar and casual, where as "my mother" is more stiff and respectful. Best to pick one tone and stick with it!

She phoned me today to say that the next Sunday there will be a little birthday party at my Grandma's house next Sunday.

This one is hard to explain but it sounds more natural when the date comes at the end! Your main focus is the birthday party and details like "at my Grandma's house" and "next Sunday" need to come afterwards. You could say "next Sunday at my Grandma's house" but the party should come before!

Also, when specifying dates, we generally don't use "the", I can't think of a reason why, just the way it works.

I saitold her that I'll go.

"I said to her" and "I told her" are both correct.
"To say" is intransitive, on its own it does not allow you to specific who you say something to.
"To say to" is transitive, it requires you to specify the person you were talking to.
"To tell" is transitive, it requires you to specify the person you were talking to.

The primerary reason is obvious (it's her birthday celebration), but I have another.

"Primary", "Secondary", "Tertiary" are cardinal words, they specify the order something occurs in.
"Primer" is a noun derived from the verb "to prime", meaning to get something ready for another more complex action about to happen. For example "the pilot primes the aircraft for takeoff."
"A primer" usually refers to a document given to someone to prepare them for some more complex information they will receive later. "I read the primer on geopolitics before watching the United Nations meeting."

I want a new haircut, and near my grandma's lives a good barbera good barber lives near my grandma's house, so I'll kill two birds with one stone.

The barber is the subject of the clause "a barber lives near my grandma", so that comes first in that clause.
While you can get away with not mentioning the house in spoken English, it's good to specify it in writing for clarity. You could also write "a good barber lives near my grandma"

Yesterday was themy mom's birthday of my mom.

AllEverything turned out well.

She liked the banana bread, but she especially loved that I did itmade it myself instead of buying one.

She ate aone of my country's traditional dish of my countryes.

She phoned me today to say that the next Sunday there willould be a little birthday party at my Ggrandma's house.

Present tense: she SAYS that next Sunday there WILL be... (present simple + future simple)
Past tense: she SAID that next Sunday there WOULD be... (past simple + conditional)

I saitold her that I'll would go.

The primerfirst reason is obvious (it's her birthday celebration), but I also have another.

"Primer" is a word, but it doesn't mean what you think–it's something you put on wood before you paint it ;)

I want a new haircut, and near my grandma's lives a good barbera good barber lives near my grandma's, so I'll kill two birds with one stone.

Good use of that idiom!