Generally you unlock pets at D.G. Level 12 and get Cat Care before Dog Care.
Dog Care tends to be more involved (more actions and upkeep) than Cat Care, so organization matters more.
XP routine (what to do every time)
Feed when available and don't let needs pile up. Play 1–2 times per short session. Clean when needed — avoids a maintenance backlog. If the hobby allows it: do training/walk in one block (10–15 min) and done.
Why is cat easier than dog?
Because dog usually has more layers (longer routine and, in many reports, more things to manage). That's not bad — it just means gains come when you do it in planned blocks.
Co-op and friend help
As with Cat Care, there are reports that friends/visitors can interact with pets (including feeding). Consider it a routine shortcut when you're focused on something else, not your main XP source.