To transfer a pet, bring it out onto your profile, then click on its image. A box will pop up with stats and at the bottom it'll say "Want to gift this pet?". Type or copy-paste the name of the user you want to send it to and hit Send Item!

Items can be transferred two ways. If you only have one or two items to transfer, go to your inventory (the brown treasure chest). Click on the item you wish to send (you'll need a pet selected to be able to do this) and then type the username in.

Alternatively, to send multiple items at once, go to your inventory and then click Quick Stock. Each item will have a row of check boxes. Check the "Send" box on all of the items you wish to send, select the quantity then type the user's name into the box at the bottom of the page.

To transfer eggs, select them in your hatchery then hit Transfer Selected and type in the username

If you have any more questions please feel free to ask :)