I just got a new pet and I want to put him into my party but I don't know how to switch out pets are put new ones in.

you go to this page
> To add a pet, scroll down to the pair of dropdown menus under the displayed pet (it shows the first pet in your stables automatically), which have the text "Use the following drop down menus search your Stables for the Pets that you wish to add to your Party." right above them.
> Left hand one is Stable Select, pick the stable that has the pet you want to put in
> right hand one is pet select
> once you find the pet you want, hit the green plus that's next to the right hand dropdown

>to remove a pet, you just hit the red minus button at the top right of their party slot

> to swap pets that are in the party, just click the two pets, like when moving pets in the stable

> hit "Set Party" when done
> if you know you will use the party again, you can Save it as a preset. If you swap to something else, then reload the preset, it automatically sets it