I love to play the piano, I've been playing since I was little.
I am not a professional, I would describe myself as a dedicated amateur who is not very good.
It is mind boggling how good some humans can play the piano (I can give many examples which you can confirm by just listening on the internet, e.g. Cziffra playing the Dante Sonata live, Hofmann playing the Chopin concertos, Horowitz playing Vallée d'Obermann).
What are ways you can learn? There are an infinite number of 'practicing methods' you can use to learn the piano, I think they are also applicable to domains not related to piano as well (e.g. programming).
Repetition
This one is the most obvious one. Repeating something makes you know it better. 'Repetition is the mother of learning'. It sounds simple enough to do, but this alone may not make the piece you're learning sound good, you may sound like 'someone who plays the same thing'. But it is the basis for most methods of learning, and not just for humans as well.
We need to repeat things to learn them, the act of doing the same thing seems to re-enforce the thing again to make the execution better on the subsequent round, and the round after that. It has a biological basis in humans (myelin sheaths get enhanced with repeated stimulation of neurons, the more myelin sheath the faster and more strong the signal between neurons, this is a very simple explanation). It maybe that repetition is a human concept, maybe nature uses it as well with evolution in some way as well? The repeated pattern of genes from lifeforms that survive are left to exist. There maybe some similarity in this but I think repetition is a human abstraction, under the hood it is a complicated thing but it a very well understood between people.
The Notes
Another obvious one, or is it? You can learn music from the notation on sheet music. The notation tells you where the notes are on the piano, what's their length in time, what's their relationship to each other. There maybe directions of how to play the notes (most music you'd expect to see this, Bach had next to nothing about how his keyboard music should be played on his manuscripts, the directions you may see on Bach music has been likely added by editors who lived after him).
How do you learn the notes?
- You could look at them...you can imagine playing on the keyboard whilst looking at them, you could be thinking how the notes would sound in relation to your hand, without the piano in front of you.
- You could try to play at the same time as reading the notes, 'sight reading'.
- You could try without the sheet music and just play on the piano.
- You could try without the music or the piano and try to visualise the notes
These were ways Josef Hofmann suggested how you could study piano in general (not just the notes).
It's easier to think of notes in groups rather than individual notes, that is easier to remember (like language, you don't remember the individual letters, you remember the words and sentences). A step up from that you could learn and commit to memory the phrases of music which might consist of many groups of notes.
The way notes are 'grouped' is quite arbitrary, the sheet music may suggest this, I think this is easier to notice/comprehend when you are looking at the music on it's own (and not have the piano at hand).
The key signature of the piece also helps in learning the notes, knowing the key signature lessens the possibilities of the next note in your mind and makes the sequence of notes more deterministic (as opposed to random notes, it helps it sound like music instead of notes or noise). Key signatures relate to each other as well, this also makes it easier to predict the next notes, the next group of notes, the next phrase.
The Rhythm
This I find difficult, I've tried many methods to tackle this. Rhythm is the arrangement of sound (and no sound) in time. All music has this, it can vary between music, it makes music sound like music (without it would sound like noise, like static from a TV).
The same methods could use as you did for the notes. A metronome is something people use (it is disliked by some as well, Josef Hofmann was very against it's use).
A method I find useful and that is difficult initially, is counting. Count out loud whilst you're playing. "one", "two", "three", "four". It is harder that it seems, you need to repeat it, once you get it, your music straight away sounds much better.
There are several other things you can do, accent a beat, make a repetitive sound (tapping your foot for example), dotted rhythm's, use different groupings of time e.g. one phrase have one set of units of time assigned to it, and a different one for another phrase. You could also try counting different numbers in the same time signature e.g. 6/8 time you could count 6 beats in a bar, or 2 beats in a bar (e.g. on quaver 1 and 4 in the bar), 3 beats in a bar (in quaver 1, 3, and 5).
There are many more. Things to do with time generate enormous amounts of approaches to tackling it (notes I think are more straight forward, as their already assigned to keys, time is something which is not easily assigned to anything).
Your strength in time becomes better with repetition (the mother of all learning). But time is a hard concept in general, it is probably poor abstraction as it means too many different things and is easy to misunderstand between humans, the ways it is recorded is of course well understood (e.g. what time is it? what's the time signature of the piece), but the execution is not as well understood between people, but certain ways to render time can sound particularly beautiful, I think any way.
Voicing
How loud each note is in relation to each other. The piano like most instruments derives it's basis from the singing voice. It tries to emulate this. It is not the same as a voice and has restrictions, it also has added abilities which a voice does not have (e.g. you can get higher range of notes, they can be played at a faster rate, they can be played together, there are more examples).
On the keyboard the range of sound can be compare to that of a particular voice, you can emulate the soprano voice, you can emulate the tenor voice.
Composers have also tried to emulate the orchestra on the piano, making piano effects to resemble instruments (Liszt is the prototypical example of the composer who has attempted this, others have followed as well in doing this).
In simple terms, you want the tune to stand-out and sound louder than the accompaniment. You want the accompaniment to sound quieter than the main tune. You might want the bass to be a little louder than the accompaniment but quieter than the main tune. You can construct 'levels of sound' which each components fits into.
There are many ways to practice these, play sections on their own, play octave apart so that higher registers are even higher and lower registers are lower, this forces you to play the lower parts quieter and the higher parts louder in order to render the piece in the desired way. Repeating this may ingrain this desire of conception (your image of the piece in your mind, you usually have one and manipulating this is an ultimate skill).
Recording yourself to sound how you hear is very valuable (recording yourself play I think is another practicing method and is a powerful one as it does not lie).
There is enormous amount about voicing which can be discussed.
Pedal
The notes and rhythm are both hard, this list of things is not in any particular order, the pedal is relatively simpler thing to implement. You need to use your 2 feet to use it (one for sustain, one for soft, if there is a sostenuto pedal then one foot as 2 pedals to use).
You could press the sustain pedal all the way down and sustain the sound. If you press this half way somewhere around this point the is a place where the notes will only sustain slightly the sound disappears rapidly. Knowing this point is useful (the half pedal), as you can activate or deactivate the sustain pedal most quickly and efficiently at this point
Another thing to note about the pedal, is that you also don't need to use it. Cziffra very sparingly used the pedal and produced tremendous effects. No pedal makes it easy to hear what you are playing. We've not talked much about hearing/listening but this is a fundamental musical skill which affects every piano playing. A lot of Bach music is played without pedal (just hear Glen Gould playing the well tempered clavier book 1 and 2).
Dynamics
Loud, soft, crescendo, decrescendo. Forte you may conceive to play loud, but it you can also consider to play strong, this works as well. You can also consider to play prominent as well. A weakness of words is the loss of nuance related to sound. Language has a tough time describing sound and emotions, it's not really designed/evolved to do this very well. It can be done and usually requires numerous words which maybe large words!.
Crescendo may not necessarily work by playing louder, although you're supposed to do that! For example you may play the main tune louder and leave the accompaniment the same volume and still get quite a good effect. Sometimes at the beginning of a crescendo you can start by going very quiet (which is noticeable in itself) and build up the sound with good effect.
You could build up the sound very slowly especially in a long crescendo. To try and lengthen it out you could find places where you can decrease the sound only to go up again whilst progressing through the crescendo (much like a jagged line which is going up), this can sometimes be effective quite effective and keep the attention of the listener.
Something to know is the receiver of the sound (the audience, the listener) should perceive there is a crescendo when they listen to the music, this does not necessarily mean you play a crescendo, your objective is that they hear one!
Tempo
This is the speed you play the piece, there maybe different tempos in the same piece. The selection of this has a lot of effect and influence in how the piece sounds. It also can influence technical challenges the piece has (having to play something fast and loud is harder than playing it slow and a little quiet).
Desired tempo can take time to achieve and can reveal blind spots when practicing. There are many practicing methods about achieving desire tempo, for example, 3, 4, 5, 7, 9, 13, 15 rhythms (playing the notes in quick succession in 3's, 4's, 5's, 7's, 9's, 13's, 15's). Linking, where you play a very small part of the piece (e.g. a small part of a phrase) very quickly, then do this for the next section, and the next section after that, then try to link the sections together by playing the small sections combined, and combining more and more small sections.
Slowly increasing speed of a metronome is another method which I think is quite an old method (this can take a long time as well), but I think is a 'practicing method', not a favoured method by Josef Hofmann who seemed to detest the metronome (his argument is if you use the metronome you'd sound like someone 'who plays like he's practiced with a metronome' which does have a sort of sound to it, he thought it produces a very mechanical sound).
Miscellaneous
Knowing a variation of a piece which is quiet and staccato I think is another practicing method (Carl Tausig reportedly did this at home after he played concerts). This may help with speed. When you try to play fast, you need to hit the notes faster, but you also need to let go of the notes faster. Hitting the notes requires you to tense some muscles whether they are in your hands, arms, shoulder (wherever that's required to play the note), letting go of the notes requires you to relax those muscles.
It's of great use to play with as little tension in muscles as possible, this allows for efficient playing and less fatigue, it also allow for fast intentional execution of notes. It is a skill which requires practice and repetition (the mother of learning) in order to get good at.
Playing in different variations of a piece is a practicing method in itself, you could play in different speeds ('slow practice', good for testing if you know the piece), playing faster than the desired tempo, playing medium speed, symmetrically inverting the piece in the opposite hand, playing it in a different key (that's a hard one, if you know the piece well this is easier to do). There are many more, you can even make these up.
There are many other methods that I've not mentioned, there are too many too name.
The most important thing is, what you play should sound nice (which is another giant infinite topic), whatever your methods are, they are a means to achieve this.