What are the advantages of training a puppy instead of waiting for him to "grow up" a little first ?

Well, many "bad" habits such as chewing on things, jumping up, nipping, not coming when called, and others, are things the puppy actually learns to do in the first few months of life. If you wait until your puppy is older to start training, all those bad  habits are much stronger and harder to get rid of than if you had started the puppy off on the right "paw". Also by training your puppy at an early age, you build a strong relationship with him that will help carry you both through the teenage months that can be difficult.