New here!  Just wondering how I can put my pets safely into my gallery, please?  Thank you!

Most users call it a Showroom :D

click the little dog next to the orange egg on the top blue bar
this will take you to the showroom
there you will see several options including the option to "stock pets" 
click that 
then use the check boxes to select the pets that you want to store then click "stock"
you can also take them out again by clicking "Edit stock"