Welcome to Subeta!

To answer your question, pets need healing only if you battle with them. If you are interested in battling, there's an entire forum dedicated to it that has lots of helpful threads to guide you. Pets do not get sick otherwise :)

Useful Links:

What to do on Subeta? Useful general information for newbies Finding friends (Chit Chat) SubetaLodge (Offsite item database, pet colours etc.)
Current Site Event: Luminaire