Your Doppelpets gain experience by fighting with you. You equip a Doppelpet by taking them out of their cage. Click on the cage icon next to the Doppelpet you want to equip.
At the end of each fight, your Doppelpet will gain 1 experience point. As long as they have at least 1 health! So keep an eye on your Doppelpet's stats during a fight.