Circle of Magic quartet by Tamora Pierce

The four books in this series (with the charming titles of Sandry's Book, Tris's Book, Daja's Book, and Briar's Book) tell the story of four children who are brought together to learn about the magic that each has. Each child has been through some hardship or another, and are all found by the mage Niko, who brings them to Winding Circle Temple. It is not until almost the end of the first book that the students realize that they actually have magic, and that that is why Niko brought them to the temple. By the end of the first book, they each have a teacher who trains them in their particular craft.
The main plotline of the four books is the story of the four teens connection to each other. In order to save themselves at the end of the first book, Sandry uses her weaving magic to weave all of their magics together. This not only enables them to learn something of each other's magics, it also makes them individually more powerful. And they become exponentially more powerful when they work together. In these stories they face an earthquake, a pirate attack, forest fires, and a plague together, using their magic and their connection to solve each problem.
Throughout the books the themes of togetherness against all odds and acceptance are prevalent. Each young person faces prejudice of one sort or another, and they all must come to terms with their differences if they are to rely on each other so closely. They grow closer at the same time as they are growing up, and growing in their power. These books are actually kind of a crossover between older children's and teen, but they could be enjoyed easily by either.