Piggies, butterflies and chickens are all hatched from seasoned eggs. You can buy butterfly, piggy and chicken eggs off the auctions, or you can use an egg seasoner to make your own once you have the necessary skills. To get plain eggs, you harvest from egg trees which are found in underground areas.
You get eggs by seasoning them. You need to have the proper skill to do this. You can also buy them in the auctions.
All eggs, whether piggy, butterfly or chicken, are incubated by the chicken. You have to stay in the area while the chicken incubates the egg. When the chicken is done, you get either a piglet, a caterpiller or a baby chick for your egg. You have to feed these babies, sometimes multiple times for them to turn into an adult, piggy, butterfly or chicken. If you want them in your yard, don't feed them until you are in your yard.
You can also purchase piglets, caterpillers and baby chicks in auction.
You can also use some youthenizing potion, and change adult chickens, piggies, and butterflies into chicks, piglets and caterpillers to cart them off to your yard. Again, feed to re-grow.