* Indigenous peoples: Native American tribes had their own systems of education, often involving oral traditions and apprenticeships, long before European colonization. These weren't schools in the European sense, but served a similar purpose of transmitting knowledge and skills.
* European colonists: The earliest schools established by European colonists were often informal, private ventures, or religious institutions. Pinpointing the very first is difficult due to limited records. Some of the earliest documented schools were associated with religious missions, aiming to convert and educate Indigenous children, or were private tutors educating the children of wealthy families. There's no single universally accepted "first".
Therefore, there's no definitive answer to your question. The history of education in America is complex and begins long before any single school building was erected.