Carrot cake and other fruit and vegetables cakes were my childhood favourites, especially if they had a good icing such as cream cheese with icing sugar. You an experiment with the fruit and vegetable mixtures you use in cakes as you become confident. Apple, banana and courgette work well on their own or with other fruit and vegetables.
If you just want to make a simple carrot cake replace the courgette (also known as zucchini) with carrot. Banana, apple and pineapple also make good additions to this recipe.