How do I "relocate" specific pets - to make "sorting" easier?

Depends what you mean by 'relocate'. If you mean within the same stable, just click on the pet picture in the stable, then click the 'stall' you want it to go in (if there is a pet in that 'stall' then the pets will switch places). If you mean moving pets from one stable to another, you click 'enable mass move' at the top of your stable, click all the pets you want to move, then choose the stable in the mass move dropdown, then click 'mass move' and all the selected pets will move to the new stable, then you can move them where you want in there using the first moving technique.

Thank you so much! Exactly what I wanted to do!