You can upgrade your pets' attack and defense levels using Pet Attack Upgrader and Pet Defense Upgrader sold by Eva Energy.
After getting these items for 100k each, you should then click on the pet that you want to upgrade. Using these items by double-clicking on them will attempt to upgrade the pet. Failured attempts don't cause the pet any harm.
Upgrading your pet gets progressively harder. While upgrading to +4 is relatively easy, upgrading to +7 may be a challenge if unlucky. Upgrading all the way to +10 is not recommended if you don't have the spare gold.
Pet drop filtering
In order to add an item to the Blacklist/Whitelist in your inventory you must open the Pet Drop Filtering tab from the new button next to the image of your pet.
A window like this will be displayed:
To add it to either list, simply drag the item to the middle box, select the tab below either of the two buttons, and click the button for the list you wish to add it to.