Pets (Cat Care) unlock around D.G. Level 12, and generally you get cat before dog.
The routine that pays off (without reinventing the wheel)
Feed when available (consistency > spam).
Do 1–2 play interactions per short session.
Clean/bathe when needed — avoids wasting time later with the pet stuck in need.
Collection / pet catalog
Once you start interacting/training, you usually unlock catalog/collection progress tied to the pet (varies by hobby stage). Same idea as other hobbies: variety and organized progress speed you up.
Co-op: do friends help?
The community says visitors can interact with pets (e.g. feed/help), which can make your daily routine easier. Treat it as a bonus, not your main XP source, since it may vary by patch/server.