Perhaps you are referring to nursery rhymes, traditional poems and songs intended for very young children. A collection was published in 1744 as Tommy Thumb's Song Book, but a 1781 compilation published as Mother Goose's Melody was more enduring, and so the term Mother Goose rhymes (poems, tales, etc.) is virtually synonymous with nursery rhymes.
The language is often archaic, and poetic English is considerably different from prose in what is permitted (owing to artistic license). As such, they are probably not useful for an adult learner as far as the mechanics of English are concerned. It is possible, in fact, that poetry will introduce confusion and non-standard usages.
On the other hand, a large proportion of the entries on Wikipedia's list of nursery rhymes are intrinsically familiar to most native speakers, and so idiomatic references abound in writing and conversation. Familiarity with characters like Ole King Cole or Humpty Dumpty or Little Miss Muffet would aid in understanding, not unlike familiarity with classical or Biblical tales or expressions.
Beyond nursery rhymes, I could not confidently say any particular poems or poets are taught to all children in all locales. To be sure, students with a decent literary education will be exposed to the works of Shakespeare, probably some Milton and Donne, and various greats in each era subsequent, but not everyone studies the same Shakespeare works nor in the same depth. Curriculum varies very considerably by location. In the U.S., it's entirely possible for high school students in two adjacent towns to study mutually exclusive sets of 19th and 20th century poets.