you have to place them in the Baby Playpens or the Barn (if you have one) and they eventually will grow up into Cow's, Horses, Deer, etc. ...

You can store most of them in the Baby Playpen or the Nursery Barn. If you store them in the Nursery Barn, there is a chance they will transform into an adult animal (not necessarily of the same kind as you put in) when you harvest the building. If you want a guaranteed adult version of the baby foal/calf/fawn, you need to click them and use the "Grow Up! (Farm Cash)" menu dialogue (this only works for certain animals).