I can't speak to the Maya, as that isn't really an area I'm familiar with (at least around the time of Contact), but in the Andes the Inca held out and fought back with numerous adaptations to Spanish weaponry and strategy.

First of all, it wasn't guns (which were slow to reload, though powerful) or armor (which was thick but bulky) that defined the Spanish military advantage. It was horses. At every opportunity the Spanish used cavalry charges to rout any attempt at reinforcement or engagement by the Inca and their allies. (As a testament to their power, the police academy I live behind in Cuzco has a big saying painted on the wall: "¡Siempre a caballo!") Atahualpa, while captured by the Spanish, marveled at these beasts, and upon his planned release (which didn't exactly pan out) he planned on taking the Spanish horses and using them for future conquests on the coast.

Now, with horses being the "big kill" - there were fewer horses than conquistadors - Inca military strategy often revolved around fighting in places where cavalry charges were not possible (restricted mountain passes, tight spaces in the cities) and disabling/killing the horse. To this specific end warriors re-emphasized the use of bolas, three stones tied to each other along a length of rope. Tying up the legs would trip up the horse and leave it and its rider open to casualty (that's not a historical picture but the principle is the same). I believe it was Pedro Pizarro who described a battle with Inca warriors, who, upon killing a Spanish horse, left the battleground for the night with the horse's head, which was mounted on a post and used as a banner the following day. More so than killing a conquistador, killing a horse was a big deal.

Other intriguing adaptations involved countering Spanish plate armor. Normally predisposed towards metallurgy as a decorative and luxury-item commodity, the Inca started producing copper arrow points, with a rhombus-shaped profile. This point could pierce Spanish steel, especially at joints but with some success in general. Certain contingents of the Anti groups - staunch Amazonian allies of the Inca and expert archers - were outfitted with these points to use in the Vilcabamba region, where the Inca rogue state of Vilcabamba remained for forty years after the murder of Atahualpa.

There are also some extreme examples of Inca warriors who used Spanish weaponry in battle. At the battle of Sachsayhuaman during the siege of Cuzco, Villac Umu left control of this fortress to a general that is named as Tupac Hualpa by Pedro Pizarro's chronicle - Tupac Hualpa was sworn to defend the fortress or die in the attempt. Tupac Hualpa was armed with an Inca axe in his shield hand and a Spanish sword in the other. Wearing a Spanish morrion helmet, he held the tallest of the three towers of Sachsayhuaman; even as the Spanish built ladders to scale the tower, soldiers would call out where Spanish were coming up, and Tupac Hualpa would overwhelm them in a flurry of swings and thrusts. Alas for the Inca, the Spanish organized a charge up the ladders and overwhelmed the captain; throwing his weapons at his enemies, Tupac Hualpa scoured his face, stuffed his throat full of earth, wrapped his head in a cloth and hurled himself over the edge of the tower, dying in service to the Inca and fulfilling his promise to die defending the fortress.

I'm also going to pass on the Maya, since the Post-Contact Maya is probably one of my weaker areas. I am not, however, aware of any notable occurrences.

As for the Aztecs though, I can saw that European weaponry was certainly adapted, or at the least captured and used. The roughly 8 month period between when the Spanish/Tlaxcalans returned to the Basin of Mexico (thus starting the actual war) and the surrender of Cuauhtemoc did not leave the same kind of time for more nuanced developments like the copper arrowheads /u/Qhapaqocha describes.

That being said, driving the Spanish out from Tenochtitlan and the Basin garnered at least some opportunities to use captured weapons. As there less than 1000 Spanish at the time, however, this meant that any sort of widescale re-armament was not practical. Nor was it required, since most Mexica forces were fighting against similarly armed Tlaxcalan (and later Texcocan, Chalcan, etc.) forces.

Still, it did happen, just not with everything. There are are no recorded instances of the Mexica forces using captured harquebuses or cannons, even though they certainly had the opportunity. The Spanish lost all of their cannon during La Noche Triste, although it's not clear whether they were captured or simply sunk (intentionally or not) into Lake Texcoco. A clearer picture of what happened to enemy cannon happened much later, during the Siege of Tenochtitlan, when the Spanish had set up a cannon on top of the main temple. The Spanish/Tlaxacalan forces were routed in that battle and "in their haste they left the gun upon the round stone of gladiatorial sacrifice. The brave warriors then seized it, forced it forward, went to drop it in the water." (Sahagún, Historia General, Bk. 12)

The explanation for this is fairly simple, the Mesoamericans had nothing equivalent to those weapons, unless you want to make an tortured analogy about bows or slings. Moreover, matchlocks and cannons require gunpowder, something that was in constant short supply. The Spanish themselves, having run out during a point in the Siege, built a catapult (which was an amusing failure). Without powder, a harquebus is just an unwieldy club and a cannon is even less useful, even in the hands of someone familiar with its action.

Crossbows were a more familiar and intuitive technology, but again, there weren't that many Spanish, and crossbowmen were a minority among them. There is an incident following a major victory on the causeways by the Mexica, wherein dozens of Spanish were captured/killed (Cortés almost among them), Díaz del Castillo does mention that the Mexica used crossbows against them. The account is a little vague as to whether the shots were fired by Mexica taught to use the weapons by captured Spaniards, or were fired by the captives themselves, but regardless, the Mexica only had 5 crossbows to work with, which made any decisive role on their part moot.

In an interesting twist on the copper arrowheads /u/Qhapaqocha mentioned, the Spanish, running low on bolts, had to adapt themselves to Mesoamerican conditions. The ordered up from craftsmen in allied towns several thousand bolts tipped with copper heads made to spec.

More efficiently and (relatively) commonly used were Spanish swords. In the most direct adaption, higher ranking Mexica soldiers simple used them in place of their macuahuitls. This wasn't simply for practical reasons, but also served both to indicate status, as well as to intimidate the Spanish. Cortés, in his 3rd Letter, states that Mexica captains thus armed in one encounter "[with] many insulting words, [were] threatening to kill us with very swords they had captured from us...."

Another use was attaching the swords to staves and using them, essentially, as spears or pikes. This was first seen by Díaz del Castillo during the attack on Xochimilco were he noted "the lances which [the Mexica/Xochimilcan forces] carried [were] made from the swords captured from us during the great slaughter on the causeways at Mexico." This can also be seen, like the more direct use of swords, as an easy analogue to a Mesoamerican weapon, the tepoztopilli, which had an edge used for slashing attacks.

Horses, like gunpowder weapons, were never used. It's not particularly clear whether any horses were captured alive, although the sources all agree that a few of them were "sacrificed." Like guns and crossbows, horses were a minority among the minority of Spanish, and were perhaps even better guarded against capture. In a counter-attack that drove a force commanded by Gonzalo de Sandoval back across the Tlacopan causeway, Díaz del Castillo specifically mentions that Sandoval "ordered us to retreat little by little so that [the Mexica] should not kill the horses." Given that there's no good indication that horses were taken alive -- the Mexica learned to target the Spanish mounts early on -- and the difficulty in learning several millennia of horsemanship in a few months, the horses actually served better as political tools. Sacrificed horse heads were arrayed, along with Spanish heads, on the skull-racks which displayed the remains of sacrifices, and a horse skull was sent, along with the flayed faces of several Spanish (beards and all!) to Mexica allies to prove the mortality of the Spanish, and thus the assurance of Mexica victory.

As for wheels, the Battle for Tenochtitlan was primarily a naval one, and the Spanish relied on the same logistic strategy as the Mexica: canoes and porters. There was innovation on the lakes as well, though it was more adaption to, rather than adoption of, Spanish tactics. The Spanish were lucky enough to have a shipwright among them, Martín López, and the manpower from Tlaxcala to cut enough lumber to make him useful. They assembled a dozen brigantines and launched them on the lakes from Texcoco. While initially tumbled (literally) by the larger ships, the Mexica retaliated by planting stakes below the surface of the water to catch the deeper draft brigantines, sinking at least one of them before the Spanish came up with techniques to efficiently break the stakes.

There was plenty of adaption on both sides, in other words. The paroxysm of violence that was the few months leading up to the fall of Tenochtitlan, however, only allowed limited "technological" adaption, and that mostly in line with already understood weaponry: swapping macuahuitls for swords, using indigenous resources to build bolts and ships. The whole war was a whirlwind of tactical innovation though, as both the Mexica and the Spanish learned to their enemy's weaknesses and strengths.

I will say that the advantage the Inca and the Aztecs had over the Maya in adapting Spanish arms was that they were single organized entities which had the time and opportunity to learn, and co-opt, the tactics and weaponry of their European opponents. While the Maya city of Tayasal did hold out until almost the 1700s and the Maya area did take decades to fully conquer, this action was carried out against disjointed opponents; the Maya were city-states, not empires. They could be conquered piecemeal, with each city having to relearn the mistakes of the one before it and with no chance to have months and months of active practice against, and with, Spanish weapons.

Finally, I'd be remiss (and /u/TasfromTAS would scowl at me in an amiably Aussie way) if I did not mention that the AskHistorians Podcast this week is Part 1 of 2 covering the Spanish Conquest. The first part ends with the Spanish first reaching Tenochtitlan, but part two gets into the circumstances that led to all the incidents covered above. It more of a broad overview, but I hear the person interviewed on the subject is quite handsome.