There's really no one answer to this because there was no uniformity in the the armors used by the Spanish. This ambiguity is not helped by the vagueness of the primary sources we have and the fact that most men would supply their own weapons and armor, supplemented with what would be supplied by the expedition leader. Gomara's biography of Cortés, for instance, does mention him spending his own money and borrowing even more to outfit his expedition with "arms," but the context makes it clear this is in reference to weapons. No mention of outfitting the expedition with armor is made and indeed armor is scarcely mentioned in the primary texts of Cortés and Díaz del Castillo, the assumption being that the intended reader (16th C. Spaniards) would already be familiar with such mundane details.
What we can glean from the primary sources (and general knowledge of the "standard" armaments of Iberian soldiery at the time) is that a helmet and gorget paired with a steel cuirass or shirt of mail would not have been so unusual. A sword and shield would be the "basic" arms used, if the person was not a specialist such as a crossbowman. Again though, individual variance would be expected, with class being a factor. The mounted troops -- who typically came from a wealthier background -- are often remarked as being more fully armored. Cortés, for instance is typically alluded to be wearing a fuller suit of armor including plate/mail for the limbs along with torso and head protection. There was also one horseman in particular that Díaz del Castillo^* calls out, Leon, who went about "clad in his coat of mail, which he scarcely ever put off."
Torso, head, and neck protection though, was more common. Since Díaz del Castillo is the most verbose and detailed on the topic, let's quote him again:
When, therefore, the attack commenced, a real shower of arrows and stones was poured upon us; the whole ground was immediately covered with heaps of lances, whose points were provided with two edges, so very sharp that they pierced through every species of cuirass, and were particularly dangerous to the lower part of the body, which was in no way protected.
That the limbs, particularly the lower limbs went unprotected is backed up by another quote from Díaz del Castillo wherein he says that, upon preparations for the assault on Mexica territory proper, Cortés:
recommended us to furnish ourselves with good weapons,helmets, gorgets, and steel coverings for the legs, to protect our bodies from the destructive weapons of the Mexicans. These precautions were, indeed, very necessary; but, notwithstanding all our defensive armour, scarcely a day past by in which the enemy did not kill or wound some of our men.
Wearing fuller suits of armor in the heat of Mexico, even highland Mexico, did have its drawbacks. Our trusty Díaz del Castillo reported that the aformentioned Leon while:
still in full armour, and had heated himself to such a degree by riding up and down, the weather besides being very hot, that the perspiration literally dropped from him, and he could scarcely breathe from over-exertion...
Thus many of the Spanish adopted the quilted cotton cuirasses (ichcahuipilli) that was the basic armor of the time in Mesoamerica which was (as would be expected) effective against the weaponry of the time and place. This was not solely a matter of comfort (and some Spanish continued to wear breastplates over Mesoamerican- or European-style cotton padding), but also stemmed from a lack of adequate supplies or ability to repair damaged steel armors.
The answer to your question, in short, is "it depends." It depends on the exact expedition about which you are talking, who in that group you are asking about, at what point in their travels you are asking about, and the availability of supplies. All of this compounded by the paucity of details from primary sources (from Mexico at least, maybe /u/Qhapaqocha has some amazing insight into Pizarro or /u/Yawarpoma has more details on other areas).
^(* I quoted from Lockhart's 1844 translation of the True History of the Conquest of New Spain for ease of cut and paste. Carrasco's 2008 translation would be my recommendation if you want to read the tale yourself.)