Racing becomes available as soon as you arrive at Skylander Academy and meet Pandergast. There are six tracks available initially (two each for Land, Sea, and Sky) and six more that are unlocked via the Land Trophy, Sea Trophy, and Sky Trophy. These three trophies also unlock Villain Cups and mirrored versions of the tracks.
To begin racing, either speak with Pandergast or select Racing from the pause menu. Check in with Pandergast whenever you are in Skylanders Academy. He offers challenges for extra rewards.