Ok, so I'm trying to put together the sections on Pet, Monster Tamer, and the Regimens to figure out how gaining and replacing pets work. If I make any mistakes, hopefully someone will correct me.
At 1st level, you gain a pet with the following limitations: CR <= 1/2, Int <= 5, type based on regimen. (Animalist = beast, Monstrous = monstrosity, Oddball = aberration, ooze, dragon).
At 1st level, this pet will follow you around, and take the Dodge action in combat by default. Its HP cannot exceed your own. As an action, you can command the pet to take the Dodge or Disengane actions, or use the Attack or Stay monster tricks. If the pet is provoked (I would assume by being attacked, but DM discretion), its default action may change from Dodge, even without the PC using an explicit action. What this default action changes to is again up to DM discretion.
I think the intent is that the PC can use the Pet befriending/turn into a pet rules from the Pet section to replace a dead pet (or simply find a better pet) even at 1st level, although the definition of a tamable creature is embedded in the Monster Tamer feature gained at level 2. My assumption is that you can only replace a pet at level 1 with the same kind of pet as defined in the Regimen section.
At level 2, you can now use Animal Handling checks on any type of tamable creature (again, CR <= 1/2, Int <= 5, type = (aberration, beast, monstrosity, ooze, dragon)). Assuming the creature is befriended and then made into a pet by Animal Handling checks, it now becomes your pet, and your previous pet is automatically dismissed. By my reading, there's nothing that restricts your choice of pet by regimen type, other than suffering disadvantage to your checks if it's not of your regimen's type (where type is implied by the Regimen but never explicitly stated).