According to Xenophon, Spartan girls received much the same physical education as boys, participating in races and trials of strength. Euripides adds wrestling to the list, while Plutarch, writing much later, includes throwing the discus and the javelin as well. They did not, however, learn to fight as their brothers did.
[Statue from Prizren or Dodona, Spartan, c. 500 BCE, source: Wikimedia Commons]
They did not neglect the things other Greek women learned either, though they spent much less time at it. A Spartan woman was responsible for running at least one household, maybe even two or more.* Whether girls learned this as part of their public education is unknown, but it was certainly important. Nor did they neglect that quintessentially female task in ancient Greece: weaving. Comparatively speaking, they spent almost no time at it but they needed to know it nonetheless.
Sparta was a society that raised its children to produce the best warriors possible. To meet that need, they trained their girls not only in how to manage a household, but also in athletics, encouraging them to compete, so that they might be the best possible mothers and the best possible Spartan women.
*If she had children with another man, for example, she had some stake in his household as well.
Xenophon, Constitution of the Lacedaimonians 1.4, 4.2 - Perseus
Euripides, Andromache 595-601 - Perseus
Plutarch, Life of Lycurgus 14.2 - Lacus Curtius
Kennell, Nigel M. The Gymnasium of Virtue: Education and Culture in Ancient Sparta. University of North Carolina Press, 1995.
Pomeroy, Sarah B. Spartan Women. New York: Oxford University Press, 2002.
Women in Ancient Sparta - Wikipedia