New pet looks: around 125 stunning new looks are now tameable in the pre-expansion patch, with nearly 190 more coming with the full Shadowlands release, later this year. There may be even more - if and when they fix a mysterious bug that prevents some newly-flagged creatures in instances (like crystal scorpions!) from being tamed.
More stable slots: if you're unsure if your stable will hold all the pets you want, have no fear! With the pre-expansion patch our stable slots have been increased from 60 to 200!
New families: there are two completely new families - Camels (including alpacas) and Mammoths (including elekks), and one semi-new family: Coursers - which are a combination of horses (new) and horned steeds (zhevras and dreamrunners, existing tames) that have been split off from Stags.
Six other families have been renamed and expanded with more models:
Dogs are now Hounds and include a number of hellhound and charhound models
Goats are now Gruffhorns and include the new ram model
Shale Spiders are now Shale Beasts and include shale worms
Silithids are now Aqiri and include kunchong and Aqir goliaths
Quilen are now Stone Hounds and include the gargons of Revendreth (coming in the full expansion)
Krolusks are now Carapids and include the gorm of Ardenweald (coming in the full expansion)
There is a new class of tameable creatures: Undead. These aren't a new family, but cover pets spread across a number of existing families. They include a number of previously-untameable deathly creatures like charhounds, skeletal raptors and skeletal devilsaurs. They also include the Undead Eagle model, which has been tameable for some time. In this pre-patch, Undead pets are only tameable by undead hunters, but when the full expansion arrives they should also be tameable by any hunter who has learned the skill from a Simple Tome of Bone-Binding, which is likely to come from Maldraxxus in Shadowlands, but we haven't yet located it.
Changes to family tranquilize/dispel abilities: in Shadowlands, all hunters have a dispel ability, since Tranquilizing Shot is being returned. This makes the similar pet family ability somewhat redundant so it has been replaced with an ability that will remove 1 disease, magic, and poison effect from the pet itself. This ability may be useful in very specific situations but otherwise doesn't seem to be one of the more useful family abilities. Eight pet families will be affected by this: Moths, Sporebats, Water Striders, Bats, Rays, Cranes, Spirit Beasts, and Stags.
Other than this, there are no core changes to families. Pet families have had no major changes and are still locked to a single spec, resulting in their usefulness being widely variable.
NPC levels have changed. With the new 'level squish', player levels now range from 1-60 and NPC levels have decreased accordingly. Almost all zones now scale from 10-15 up to 30-45, depending on the expansion.
Taming level requirements: with the new scaling system hunters can tame non-elite pets up to two levels higher than them. Elite pets require you to be the same level or higher. There are a number of elites in older content that won't be tameable until the full expansion is released and you can get to level 51 or 52.
Exotic Beasts is now a level 39 skill, which is roughly equivalent to level 90 in old terms, compared to 65 in BfA. Disappointingly this means most spirit beasts cannot be tamed while leveling in their zones, and a number of exotic creatures in starting zones can never be tamed.