A boy named Elmer Elevator has adventures with a young dragon (who is named Boris, although that isn’t mentioned in the first book) that he rescues from an island of wild animals. All of the animals in the books speak in understandable human language. Elmer originally heard about the captive dragon from a cat he befriended but his mother wouldn’t let him keep. After Elmer leaves to go rescue the dragon, the mother has a change of heart and takes in the cat. The cat lives with the Elevator family through the rest of the series.
The books in the series are fairly short, less than 100 pages each, and they’re fairly easy to read. The first book in the trilogy is a Newbery Honor Book.
The books were written by Ruth Stiles Gannett and illustrated by her stepmother, Ruth Chrisman Gannett. All three of them are available to borrow and read for free online through Internet Archive (most have multiple copies).
Books in the Series

My Father’s Dragon (1948)
A boy named Elmer rescues a baby dragon from an island of wild animals.
Elmer and the Dragon (1950)
Elmer’s continuing adventures with the young dragon and his return home to his parents and cat.
The Dragons of Blueland (1951)
Boris the dragon returns to his family, but he needs Elmer’s help to protect them from men who are trying to catch them.