You can trade to pets directly using the standard trade system. This system can be accessed in one of two ways:
Trading Page 1 of 2
The first page of the trading screen is used to select the things you wish to trade. To select items use click and drag on the left window this may sound like an odd way of doing it but it means you can select a bunch of items quickly without having to take your finger off the mouse button which makes it a pretty quick and intuitive way of doing things.
You know when an item is selected as its background color will change to indicate that fact.
You can also use the buttons in the middle of the screen to select items in bulk:
On the right side of the screen you have a box that allows you to enter an amount of money which will be given along with the items to the person you plan on trading with.
If you wish to trade items from a bag other than the default you can use the bag buttons at the bottom of the screen to switch back and forth between bags.
You can only trade items from the same page of your inventory in one go so if you plan on trading items across multiple inventory pages you may need to trade several times.
To proceed to the next screen click the Next button in the center column of the screen.
Trading Page 2 of 2
The second page of the trading screen is used to confirm the items and money you are trading and also presents you with a quick confirmation on the target of the trade.
To go back to the previous page to make a change click the Previous button at the bottom of the screen otherwise click the Complete button to finish this trade and give the items and money to the selected pet.