A note on the staff shows you which key to play, but rhythm shows you when to play it and for how long. Rhythm can make a melody sound off even when the notes are correct. This is because the fingers have found their targets, but the beat has not told them where to rest.
Before touching the piano, look at a new system. Look for bar lines, rests, and note shapes. The quarter, half, and whole notes do not require the same length of time. If you do not check the length before playing, the hands may start guessing what the rhythm is. Counting is important because it establishes an anchor; the rhythm keeps steady until your fingers arrive.
Count the rhythm before pressing the keys, using as simple a system as the music calls for. You can say “one, two, three, four” for each measure and point at the notes while you count aloud. If there is a rest, do not stop counting; silence is part of the rhythm, and you may want to count through it anyway. It is not playing if you miss the beat.
While you count, tap it on your leg with your hand. Tap the beat first, then tap the rhythm of the notes. This is a way of tapping away from the piano. It keeps the beat separate from the notes so that you can focus only on the rhythm. While you do not play, you do not need to think about which finger to use, whether you should curve the fingers, or what C is. You should just notice if the rhythm moves evenly across every measure.
Count first, then play. Pick out one or two measures, position the fingers on the keys, and count aloud. Then play the notes while counting aloud. Keep the count slow enough to stay on beat with the music. If you count too fast or the notes run ahead of the count, go back to the tapping. Rhythm takes precedence.
A metronome should be used, but at a slow setting. Turn it off if the clicks make you feel stressed. It is more helpful to count your beats out loud or tap your beats before you play. If it is necessary, you can count the notes and play along with the metronome. The metronome is not there to make you feel hurried; it is there to ensure your count is consistent.
There is only one sign of improvement to look for, and it is simple: the measure is predictable. Before you press a key you know what will come next. You can let a note go for its full length, and you can wait through a rest with your fingers in place. You can finish a system or start a new one without a scramble. If the rhythm is counted before the keys are pressed, the music may feel less like a line of surprises and more like music being played.