If I told you I had a magic pill that would improve your rate of progress at the piano, you'd want to know about it, right? And if it had no down-side or unwanted side effects, you'd almost certainly want to take it.
I do have such a pill. It's called "playing without looking at your hands."
I've talked about this in previous posts, but I want to emphasize again how important this is. I'm speaking largely to beginner and intermediate students; advanced pianists almost certainly already do this.
Playing without looking at your hands helps you develop your kinesthetic sense. I call it "knowing the geography of the keyboard." When you look at your hands, you bypass the opportunity to develop the kinesthetic. Anyone who has seen a great pianist, or even just a good one, can see that the act of playing the piano is very physical, and that physical ability must be acquired though excellent training, just as it would for an athlete.
When you start out as a beginner, you probably will use a book for beginners to learn to read music. These books all start you in a "five-finger position," meaning each hand is over just five notes, so everything you need to play will be right under your hands, so to speak. Even using this very limited range of notes, there are still plenty of challenges as you develop the coordination you need, not to mention reading rhythm correctly, etc. The book should be emphasizing reading by interval, so that your hand learns to navigate by interval, meaning it feels the distance between notes. If you first learned to read, whether by teaching yourself, or with a bad teacher, then you may have learned to read using note names (E,G,B,D,F etc.) then you very likely started looking at your hands right from the start, because, presumably, if you wan to find G, you'd need to look down at the keys to see where G is. This is a terrible way to start. If you can't master just a five-note range without looking, you will find it extremely difficult to move on to more complex and more interesting music. You may eventually play the more complex music, but it will take you MUCH longer to learn and master, because of the constant looking back and forth between the score and the keyboard, but mostly because you just aren't developing the kinesthetic skills you'll need. In more complex music it will be impossible to look at both hands at once because of where they are on the keyboard. Trying to search and find every note by eye is far too slow, except maybe for simple -- and slow -- music.
Soon the books will expand the range to six or maybe eight notes in each hand. Then it will have you play pieces where it is necessary to move your hand(s) to a new position, that it, to different notes. The books won't have you doing large jumps (that comes later), but you can't just "cling" to the five or six notes where you started. The sooner you do this the better, because many students like to cling to that five-finger hand position -- it becomes a kind of security blanket.
When you move to each new challenge, the urge to look at your hands becomes very strong. It is all about the fear of wrong notes. Yes, in learning to play the piano, you will play many, many wrong notes. Instead of cringing, getting frustrated, and immediately wanting to look at your hands, you must recognize this is part of the learning process. A figure skater, in learning to do those amazing jumps, is going to fall a great number of times. If a skating student said "but I just don't want to fall!" they will never become an advanced skater. So it is with piano and wrong notes.
Not looking at your hands has other benefits as well. When you hear a wrong note, you want to be able to identify by ear what the problem is, so you can adjust and move on. So your auditory sense will also get stronger. When you hear a wrong note, if you always, stop, look, try to "correct" it before moving on, this WILL become your habit. You can't just think that one day you will suddenly play with fluidity if this is how you have practiced.
I should mention that if you are playing by ear, or improvising, or playing a piece that you have memorized, you don't need to stare off into space. You will look at the keyboard just to keep a general eye on things, but certainly not looking for every note. You can't imagine that a great jazz improvisor is looking at his hands to see what notes to play!
I believe that playing without looking at your hands is one of the most important keys to unlock the ability to make the progress. Unfortunately, many people find my "magic pill" difficult to swallow. Nevertheless, I urge you to persevere. Looking at your hands to find the correct notes may seem like an obvious solution in the moment -- you may think, hey, aren't we supposed to play the right notes?? Getting the right note because you looked at the keyboard to find it is equivalent to winning the battle, but I assure you, you will lose the war. You may get the immediate gratification from the "correct note," but you are hampering your long term success."