Papaya was one of the first fruits the twins had once I began venturing beyond commercial baby food. After letting the papaya ripen for a few days, I remove the skin (some papayas have a thick skin i.e. the slight bitter after-taste that is just under the skin) and seeds. Half of it I puree and food cube. The other half gets cut into toddler size pieces.
The puree I thaw for breakfast and the pieces make a great snack rolled in graham cracker crumbs/powder. Some papayas will have a bitter after-taste no matter how many days I let them ripen. The best ones, in my experience, are from Whole Foods and my local Indian and Asian grocery stores.