Question: I know pretty well that the plural for 'person' is 'people'. But my literature professor used once the word 'persons' because, he said, he was using the word the same as it will be used 'individuals'. Or at least I understood it that way. 

So, my question is: Can I use the word 'persons' in the next phrase? (I wrote the whole phrase in order for you to have the correct context.)

> 
> I'm so into the idea of developing each skill alike, the idea of being
>  round **persons**.
> 
> 
> 

Is it okay, or should I just use 'individuals'?

Thank you! :)

Answer: 
Your sentence doesn't quite make sense and I think you need to use "individuals" if you want to refer to more than one person.

You could say:

> 
> I'm so into the idea of developing each skill alike, the idea of being
>  a rounded person.
> 
> 
> 

or

> 
> I'm so into the idea of developing each skill alike, the idea of having
>  rounded individuals. 
> 
> 
> 

You can't really use "being" with "rounded individuals" as you can't "be" more than one person.

Also, you don't say what audience this is meant for.

"I'm so into" is a very "hip" was of saying "I am very enthusiastic about". For a more formal situation you probably should use the latter.
I disagree with your premise. "Person" is a noun that identifies a single individual human being. "People", on the other hand, is a noun used for a collection consisting of "persons". "People" seems like a plural for "person" because in order to have a collection, one must necessarily have more than one. Because in many instances when one refers to multiple individuals it is not necessary whether to specify whether one is referring to the group or the individuals (three people standing on a street corner can be as accurately described as either "those three people over there" or as "those three persons over there"), people gets more usage than "persons", but people is not the plural of person. "Persons" is the word to use to specify more than one individual, when it is necessary or desirable to retain the reference to individuals.