Cortés brought only horses with him to Mexico. In Cortés 1st Letter (ostensibly not written by him, but by his handpicked council of the newly founded city of Vera Cruz) he left Cuba with "10 ships and 400 fighting men... 17 being mounted." There's some variation, with his secretary Gómara writing later that it was 11 ships, with 550 men, 50 of whom were sailors, and 16 horses, along with "200 islanders from Cuba to act as bearers and servants...."

Later arrivals of Spanish, who were to either arrest Cortés or resupply him, similarly brought nothing but men and horses, with often unnamed and unnumbered amounts of porters. There are reasons for this. These early expeditions, which was basically coastal exploration and trade. The early Spanish did not expect, and were expected, to venture far from their ships, and neither of two expeditions before Cortés went further inland than was required to find fresh water. Cortés' group only trekked overland after confirming there was a rich and powerful "Kingdom of Mexico" further inland; founding a town as a legal loophole to get him out of his obligations to the governor of Cuba; and allying with the large and populous city of Cempohuallan. That same allied city also supplied Cortés with a number of professional porters (pl. tlamemeque in Nahuatl) to accompany the Spanish on their journey, thus initiating what would be crucial role of the Spanish relying on the Natives for logistical support (and later to make up the bulk of the troops).

There weren't a lot of pack animals in the early voyages because cattle, donkeys, mules, and the rest are expensive, difficult, and dangerous to transport. The horses made up a vital part of the military force, so it made sense to bring them, but they had to be transported using special slings which kept them from toppling over and injuring themselves or other during the sea voyage; other livestock could pose similar risks to themselves and others on the waves. While livestock can also be turned loose to graze on land, a cow on a ship still has an appetite like a cow, both for feed and, more vitally, for water. You would not bring livestock with you on journeys into the unknown or to where you had no intention on permanent settlement. This was part of the reason for the popularity of pigs with later Spanish expeditions: they were less prone to injurious falls; a pig stampede is less terrifying than one of cattle; and have a more catholic diet than finicky ruminants.

(As a total side note, there's an incredible legacy of Spanish-Swine exploration on of Georgia's Sea Islands, Ossabaw. The 4+ centuries of separation from other pigs have led to a rather unique breed of swine, which has adapted to the boom-bust bounty of acorns to essentially stuff themselves in Fall then starve the rest of the year. This had led to their use in diabetes research and, when crossbred with regular hog breeds, some delicious eating. As an even bigger side note, if you're ever in or near Atlanta, this place carries absolutely mouthwatering ossabaw pig meat.)

When the Spanish did start to permanent settle in Mexico however, they very rapidly began to transport livestock to the mainland. That ranching had already been established in the Caribbean meant the hard labor of getting the animals from Europe to the Americas had already been done; part of the wealth Cortés used to finance his expedition came from his cattle ranches in Cuba. The venerable and seminal Gibson (1964 discusses how many of the early land grants to would-be-hidalgos coming over to Mexico were aimed at providing pastureland. He writes of the exploding population of livestock:

The earliest grants proved insufficient to contain the expanding herds after only a few generations. Herds of sheep doubled in size in a year and a half or less. Herds of 12,000 to 15,000 head are recorded as early as the mid-sixteenth century, and of 50,000 to 70,000 in the late sixteenth and early seventeenth centuries. The hacienda of Santa Lucía is said to have had 140,000 sheep and goats, 5,000 horses, and 10,000 cows and oxen in the late seventeenth century.

In other words, when the Spanish began to permanently settle in Mexico, they did there best to rapidly create the conditions to which they were either accustomed to, or to which they wish to be accustomed.

If this sounds incredible disruptive to the indigenous landscape and peoples, you would be correct. Gibson goes on to say:

Indian communities constructed corrals and fences and trenches to control wandering Spanish cattle and to surround and protect agricultural lands. Indians fought back with nets and traps and pits and dogs and fire. But in general cattle were more successful in finding forage than were Indians in protecting their crops.