There are a few really obvious things I've learned about piano playing. They weren't all obvious when I began.
The level you need to get a piece to to play for anyone else is much greater than the ability you have when you think you can play it for yourself at home. First, there is the added nervousness; secondly, you aren't really playing it at home as well as think in the first place.
To play a piece well, it has to be far easier than your intermediate level. It has to be something that you can play without any hesitation for the notes. All the attention most be on nuances: dynamics and exact phrasings. This applies to difficult pieces as well. They must seem as easy as a simple child's piece in order to sound good.
Muscle memory and proprioception rules. It is not sufficient, because you can blank out mentally and the muscles won't do what they are supposed to do. You can also engrave mistakes into your muscle memory. Mistakes are not accidents, but things you have learned wrong. You should know where your hands and figures are, always. Looking at the keyboard is not as efficient.
Fingering has to be efficient and consistent. It is hard to have good muscle memory if the fingering varies each time you play something.
Looking at the written music, by the same token, is distracting from actually playing. You have to get to the level where you either know the piece by memory, and the score is only a reminder, or else be so good at reading the score that there is no difficulty there. Imagine trying to recite a poem you didn't know in an alphabet you could barely read. You would either have to know the poem well, and use the alphabet only as a crutch, or get better at reading that alphabet.
Linear, right hand patterns are not difficult, inherently. They can be learned slowly, and gradually sped up. The same goes for linear patterns in the left hand, involving only one note at a time. Most of my difficulty seems to come from playing more than one note at a time.
Even numbered intervals will be written on a line and a space, so a fourth, a sixth, and octave, etc... Odd numbered intervals are written all on lines or spaces, so thirds, ninths, etc... This pattern extends as far as you can count. Intervals, then, look a certain way on the page, and are more distinctive looking than individual note values.
The placement of the note C on the piano is symmetrical. So it is the next to the bottom space on the bottom, the next to the top space on the top; two ledger lines above and below the treble and bass clefs.
Every note will change from a line to a space or space to line at every octave.