I teach students from age five and up. From my experience, the most challenging aspect of learning to play piano is the hand coordination. Getting both hands to do two different things at the same time just takes practice. I never have a problem getting students to understand music theory concepts, but the hand coordination just takes time and careful practice. Having a PhD in math will not help with this at all. When my more experienced students get frustrated with this, I take them back to practicing extremely slow until they develop some muscle memory with what they are attempting to play.
The first step I recommend when learning how to play a piano, is getting familiar with the keyboard layout; there are actually only twelve notes that repeat up and down the piano. Once you understand that concept you can begin to formulate a perspective about the piano that will allow you to simplify patterns and particular fingerings that move you up and down the keyboard. It is important to develop this perspective from the beginning and it will make life easier and help you to learn faster. You will find out that learning to play the piano, or any other instrument, is an accumulation of a lot of small concepts and disciplines that snowball into being a skillful musician.
Most of your practice and study in the beginning is to develop a familiarity with the keyboard, your body, and your mind. From the time you learn the names of the keys and your first scale you are setting up a visual in your mind and developing a physical interaction with your hands and your entire body. The way you sit and hold your hands matters; the way you hold your head to one side or the other matters; the foot you use to keep timing matters; and the way you breathe matters. When you first start learning to play piano, it is good to have a teacher because they will point out all these details you may not be aware of as a beginner.
I find it very helpful to break the process down into several different categories. The first category is music theory (how music works), and this is mainly centered on scales and chords. I like to start with chord construction and work into learning major and minor scales in different keys. The second category is the physical aspect of using the proper fingers and building accuracy and speed. I have noticed that typical fingerings that feel natural to me can feel very awkward to less experienced players. So I always encourage students to practice slow and teach your fingers new things. It is very easy to have your fingers dictating what you play when your mind should be dictating to your fingers what to play. This is one of those subtle things you must constantly be aware of. I remember meeting a piano player years ago who played everything primarily on the black keys. I asked him why he always played in difficult keys and didn’t play in the keys that are most familiar to everybody, like the key of C. His reply was, he was learning to play piano on an old broken down piano and most of the white keys didn’t work. It was actually difficult for this guy to play in the key of C, which is all the white keys and where most people first learn to play. His fingers were so accustomed to playing primarily on the black keys, it felt natural and that’s where he stayed. The next category is actually reading music. The first step in this category is learning the lines and spaces on the staff. I suggest you learn these like the back of your hand. As you are getting familiar with the lines and spaces, you will learn how to read rhythms. If you are serious about learning how to play piano, this will be an ongoing process. It starts slow with quarter notes, half notes, and whole notes, and it will gradually get more complex until you reach a point where you are able to subdivide and decipher any rhythm you encounter.
There is another category of study I believe that should be developed and practiced even more intently as the other categories. This category is ear training, and some people believe you have this gift or you don’t. Some people are born color blind and not likely to become professional artist, but not impossible. Some people are born tone deaf and limited in how they hear pitches, and on the other extreme, some people are born with perfect pitch; most of us fall somewhere in between these two extremes. If you practice developing your ear, you will be amazed at what you will be able to hear and reproduce.
When learning to play the piano, if you start with the right approach and focus your time and study on these categories, your progress will be phenomenal. Keep in mind there are always some up and downs throughout the process. The key is to always search out something new to keep your musical adventure fresh. Don’t just practice the mechanics all the time. Play with other musicians, learn new songs, improvise, and most of all have fun.