It is not the purpose of this FAQ to tell you how to locate a pet, but you should be able to figure this out initially by looking at the Pet Status Screen (via the top right hand drop down in game).
Hint- look for an NPC called Jenny. Use the Finder located on the footer.
As and when time permits more pet options will be added so that gaining pets will be possible from various occupations and methods. At the time of release there is only one method open.
Of course you can ask your fellow captains in the forums, or chat with them in our Discord Chat channel.