Yes, taking a protein shake for breakfast can be a good idea. It will provide you with the amino acids to switch your body into an anabolic, muscle-building state, while also spiking your insulin levels so that you have more energy throughout the morning. It is also an easy meal to digest. Many people find it hard to digest food first thing in the morning. A protein shake is also a good choice as a pre-workout drink if you are heading to the gym first thing in the morning when you don't have time for breakfast foods.