Carrots are by far the most efficient source of energy and XP for bunnies, but not for your level XP. They take 30 seconds to grow and provide 1 XP when picked (for a multiple of 1 minute to 2 XP), as opposed to Green Leaf Lettuce which takes 1 minute to grow and gives 4 XP when picked.
However, it still provides the best source of energy for bunnies: 30 seconds for 2 energy as opposed to radish, which are 5 minutes for 3 energy. Also from the store: 50 gems for 2 energy, versus celery: 50 energy which is ten times higher, but the price is fifteen times higher.