Machine vs. Human
Part of learning the piano is simply a battle of Machine vs. Human. The piano is a machine like no other, both in complexity and difficulty. If you’re an adult trying to learn the piano, the first hurdle is pure, brute strength. In order to wiggle your fingers repeatedly and accurately, you will need to take the approach of the athlete or gymnast.
Your hands are the last extension of the great machine that is your body. No amount of brilliant mental management will make up for weak hand and fingers. No amount of brute strength will make up for dull mental management. Playing the piano at a high level takes training similar to athletes and ballet dancers: it's a lot of work.
Weak Fingers Make Difficulty
Beginning adult pianists often think they are mentally deficient when a difficult passage makes them falter. But it is most often not your mind that falters, but your fingers. You’d be surprised how many times you stumble at the piano because of weakness rather than mental disorganization or fuzziness. So if you want to take greater control of the piano as your hobby, start training your fingers in equal measure to how much time you spend simply playing.
Start With A Warm Up
If you normally play about 20 minutes, try a combination of ten minutes exercises and ten minutes playing. Do the exercises as a warm-up first, just as if you were going to the gym. With this regimen, you will be surprised at the strength in your fingers, which will begin to appear after a few weeks. Yes, that does seem a long time, but it is worth it.
One way to survive the dull repetition of finger exercises is to watch a TV show while you play. It doesn’t take much of your brain to watch the TV. Your fingers will wiggle along by themselves just fine without your brain's intense involvement.
The first exercises to try are the Virtuoso Pianist by Hanon. Try the first one and do it over and over. Start with just the right hand. Then do the left hand, each hand separately. Lift the fingers high, and take a full stroke (with most people this is about 2-3 inches above the keys.) If you feel tired or feel any pain in one hand, stop and do the other hand till the discomfort subsides. If you feel tiredness or pain in either hand, stop and come back later.
Never “play through” pain in piano exercises. The muscles you are using are usually completely unused by non-piano life, and will be extremely weak. Like an athlete, you must build them up, slowly and without injury. Think of your fingers as baby fawns or calves, weak in the legs and barely able to stand: treat them gently at first. Your body is a machine, and so is the piano with 9000 moving parts. Rather than blame your brain for mistakes, start training. See how much the problem was simply weak, unused piano finger muscles.