"I HAVE already goGONE home.

The sentence "I ALREADY GO HOME" does not have a correct tense structure. That is why you should add "have" after "I" to form the Present Perfect tense. The correct sentence would be "I have already gone home."
If you add "am" after "I," the sentence still would not be grammatically correct because auxiliary verbs BE (am, is, are, was, were) cannot be followed by a base verb (Verb 1). However, if you intend to use the Present Continuous tense, you should change the sentence to "I am already going home."

I'm AM very glad that coI came today because THE GAME was so funny game.

When creating sentences in English, every verb must have a subject before it. However, in your writing, after using complex conjunctions (such as "because" and "that"), there is no subject, which makes the sentence confusing. Therefore, we can change it to "THE GAME WAS SO FUNNY" and "I CAME TODAY." From a grammatical perspective, you must use the past tense (Verb 2, come → came) because you are referring to "today" as a past event that has already happened.

When I will go home, I'll WILL cook a dish.

In time clauses like “when,” we don't use “will.” Maybe, if you try to practice English writing, you should not use abbreviation sentences because it is not formal if you take an English exam

Maybe I'll cook a porridge of carrotCARROT PORRIDGE with sugar!

“Porridge” is an uncountable noun, so no ‘a’ is needed.

After that, I'll WILL message with Diana and my parents and will have a rest.

The verb "message" does not require "with." Additionally, there is no need to repeat "will" before "have a rest." Moreover, you must add a comma after using an adverbial phrase such as "after that"

Tomorrow, I'll go on theplay volleyball at fifteen3 p.m., and I hope that I'll play tomorrow same likeo play as well as I did today."

"I al'm ready to go home.

"already" would suggest you are at home. Also needs verb "to be".

I'm very glad that coI came today because was souch a funny game.

"came" as it happened in the past. Need indefinite article in front of "funny game".

When I will go home I'll cook a dish.

No "will". Implies more of an intention.

Maybe I'll cook a porridge ofoatmeal with carrot withand sugar!

I'm not sure what it is you're cooking so I've put something in English roughly fits the description.

After that I'll message with Diana and my parents and will have a rest.

"message someone" no "with". "I'll" is short for "I will" so you don't need the "will" in "will have a rest".

Tomorrow I'll go on theplay volleyball at fiftthreen p.m. and I hope that I'll play tomorrow same likelike I did today."

"three p.m." or "fifteen hundred" or "fifteen hundred hours". "three p.m." would be more natural sounding and the most frequent. p.m. is an abbreviation; it is short for post meridian so both letters have a "." after them.

"I alm ready to go home.

Or you might mean, "I am already at home"

I'm very glad that coI came today, because it was souch a funny game.

Past tense of come = came

When I will go home, I'll cook a dish.

Alternative= "I'll cook dinner"

Maybe, I'll cook acarrot porridge of carrot with sugar!

I've never heard of carrot porridge, so I'm not sure if this is the right word or not

After that, I'll message with Diana and my parents and withen I'll have a rest.

Alternative= "then I'll go to bed."

Tomorrow, I'll go on theplay volleyball at fifteen3 p.m and I hope that I'll play tomorrowhe same likeas today."

If using the 24 hour clock then you don't need to use "am/pm". Personally, where I live, we don't use the 24 hour clock