The Aztec's conquered like any other Empire. They sent soldiers in, fought the locals, defeated them, and exacted tribute after winning. The process went something like this: First, the Aztecs sent emissaries to their intended target with a demand to surrender and accept a tribute deal. This was often framed as a request. The Aztecs would say they were building a temple, and wanted their target to contribute. However, there were other causes to attack. The Aztecs relied heavily on professional merchants called Pochteca, who travelled to distant regions to procure luxury goods such as feathers and cacao. However, this made them targets (plus, they were often, correctly, identified as spies). Many Aztec attacks were provoked by the harassment, or even murder, of their pochteca.

When the Aztec army actually arrived they wanted to fight the enemy in the field. Therefore they were not too concerned about an enemy knowing they were coming. Partly this was because the Aztecs wanted to defeat the enemy very 'publicly,' demonstrating their valour and strength for all to see (and be intimidated by). However, their approach also exacerbated political divisions within their target city. Their enemies were given another chance to surrender and accept a tribute arrangement. Plus, the Aztecs preferred to fight outside of cities if possible.

If the battle commenced, both the Aztecs and their rivals would fight to defeat each other as viciously as possible, using a variety of tactics to break each other's moral and cohesion. These fights could be very bloody, with thousands of casualties. When the enemy broke warriors would surge fourth, killing enemies and seizing captives for sacrifice. After winning, the Aztecs would give their enemies one last chance to surrender and accept a tribute deal. If refused, the warriors were given permission to sack and loot the city.

Obviously though, this is something of an idealised narrative. Nevertheless it covers the key elements of Aztec imperialism. The idea that Aztec Warfare was especially ritualistic comes from confusing regular warfare with the Flower Wars, which were purportedly much more overtly religious. Although several modern historians such as Ross Hassig and Barry Issac have pointed out that even the Flower Wars were highly political, and were intended to exhaust Tlaxcala through prolonged low-intensity warfare.

As for the Spanish, I'd say no significant effect. The Aztecs were initially hampered fighting the Spanish, because they initially did not realise what their objectives were, allowing the Conquistadors to get the drop on them. Furthermore, in the early battles their tactics were aimed at fighting a local rival, not a European warband, and so they had difficulty dealing with Cortes' forces. The Aztecs responded to these problems like anyone else would. They adopted new weapons and tactics. For example, to fight cavalry they built lances. They also armoured their canoes to protect their warriors from gunfire and crossbow bolts. They also began to use more night attacks, ambushes, and field fortifications. In the end, the Aztecs were exhausted from hunger, thirst, and attrition, rather than defeated by Spanish arms.

Sources:

Aztec Warfare: Imperial Expansion and Political Control by Ross Hassig

Aztec Warfare: Goals and Battlefield Comportment by Barry Issac