You can adopt pets from Personal > Pets > Create Pet accessable from the top nav bar, or visit the Adoption Centre at Delphi Beach (Under Subeta > Explore on the top nav bar)

Useful Links:

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