I don't think "I and Julio went to the schoolyard." is grammatically incorrect.
It is simply not the common (and polite) form.
See for instance the personnal pronoun section on compounding:
When a personal pronoun is connected by a conjunction to another noun or pronoun, its case does not change.
We would write:
I am taking a course in Asian history"; if Talitha is also taking that course, we would write:
Talitha and I are taking a course in Asian history."
(Notice that Talitha gets listed before "I" does. This is one of the few ways in which English is a "polite" language.)
The same is true when the object form is called for:
Professor Vendetti gave all her books to me"; if Talitha also received some books, we'd write:
Professor Vendetti gave all her books to Talitha and me."