Parallel with the study of piano is the need to train your musical ear, so that you develop a good relation between what you hear internally and which notes your fingers are playing. While our ears do become better just from our experience of practicing piano, most of us can benefit from supplementing this with specific exercises designed to improve our "musical ear."
Some of the ways we can do this include: figuring out melodies and chord progressions "by ear," singing arpeggios using a neutral syllable such as "la," or going through a solfegé book (do-re-mi, etc.)
Ear training is a longterm process, and the results are well worth the effort. Whether you play jazz, rock, blues, pop, or even classical music, your playing will improve as you develop your ear. You'll also experience music in a fuller and more intense way. The notes in the middle of your chords will begin to "pop out" at you and you'll be able to take in a more detailed picture of the music you listen to. Also, the better your musical ear becomes, the easier it will be for you to memorize music since you'll be able to simply play what you remember. And as an improviser, your solos will be a more direct expression of who you are as a person, since there will be a more direct connection between the music you imagine and what comes out on the piano or keyboard. You'll also begin to compose music more rapidly, since you'll be able to play the melodies you imagine quicker without having to spend a lot of time figuring them out on the piano.
But be persistent and don't get discouraged if you don't see "improvement" in the same way you do when learning to play a piece on piano. In this way, ear training is akin to learning a sport. You'll see some improvement from week-to-week, but persistent practicing will yield the biggest results over a time period ranging from a few months to a year. Start training your ear now, and you'll be very happy when you begin hearing music more clearly. Your musical confidence will begin to grow when you know how the sounds you hear are translated on the keyboard.
Ear training is a journey that all successful musicians take at one time or another. Start now, and make sure you enjoy every step of the way!
Whether you're just getting started with piano improv or need to take your current playing to the next level, here are some free lessons you'll enjoy. Have fun!