It depends on the period you're interested in. For the time of the rise of Sparta, in the 6th century BC, there are no primary sources describing the Spartan upbringing. The first comprehensive look at their education system and social organisation is Xenophon's Constitution of the Lakedaimonians, which dates to the middle of the 4th century BC, and describes the system probably in place from the middle of the Classical period. You can find the text here.

Later on in the 4th century BC there's quite a bit of discussion of Sparta as a political and social ideal; we learn much about it from Plato's Laws and Aristotle's Politics in particular.

Later sources are more difficult to use. They represent the end point of a long tradition mythologising Sparta and its supposed early laws. The version of Sparta sketched in Plutarch's Life of Lykourgos and his various other works on Sparta is clearly corrupted by the propaganda of the 3rd century BC reformer kings Agis and Kleomenes, among other things. If any of it is accurate, it describes Sparta as it became during the Hellenistic period, when the upbringing became stricter and more ruthless. Its elements should not be projected back into the Classical period, let alone into the Archaic period, for which our sources are almost entirely lacking.

For an excellent summary of recent scholarship on ancient Sparta, including discussion of what sources we can and can't rely on, have a look at Stephen Hodkinson's 'Transforming Sparta' (2014).