Everyone should be equally or should be equality. Which one is correct ?

Look them up in the dictionary.

In "Everyone should be _________," you're modifying "Everyone." "Everyone" is a pronoun, so the thing that goes in the space needs to be an adjective.

"Equally" is an adverb. So that can't be right.
"Equality" is a noun, so that cant be right either.

Look around in that area of the dictionary for an adjective that means what you want.

And have you ever played "Mad Libs"? If you play with someone who knows the parts of speech and you'll learn them pretty fast too. No, it's not as entertaining for little kids (I was introduced to the game by a 3rd grader at a time when I hadn't done much beyond nouns and pronouns), but it's better than rote memorization and it works.

Good luck mastering all this. It will really help your writing.

Neither.

Equally is an adverb.
Equality is a noun.

You need an adjective in that space.
Everyone should be equal.
Everyone should be treated equally.
Everyone should be treated with equality.

Everyone should be treated equally under the law.
Every should be viewed as equal in the sight of the law.
There should be equality in the the administration of justice.