Of all the things I stress while teaching piano lessons, the most important is proper technique. This starts from the very first lesson with my beginners and is so important as to often merit not advancing in skill level and complexity so we can attain proper hand and arm form.
(Before I go on: My apologies, it's been awhile since I've written an original advice post—February 6.)
If you're wondering what proper form entails, it's actually very simple: it's natural and relaxed, meaning your arm, wrist, hand and fingers are meant to be relaxed. To illustrate what hand relaxation should actually look like, you need to know what it should not look and feel like first: put your hands up and extend your fingers out all the way. Stretch your finger tips out as far apart as possible. This is a strained hand and is requiring dozens of muscles, putting them to work to such an extent that if you keep doing this for some time, those muscles will fatigue. Much like a frown using more muscles than a smile, the proper hand form should actually utilize fewer. In fact it often feels wrong at first because it is, mechanically and technically, easier.
Are you still hanging out with your fingers extended? They'll probably hurt then! Completely relax your fingers (if you can't or don't know how, then shake your hands out) and let your fingers go limp. Your fingers should be naturally curved like they are when they're at your sides while you walk. Now extend your hands and fingers again while closing your eyes. Did you feel each muscle tighten? Do you feel the little muscles surrounding the bones on each finger? As you stretch out your fingers your palms flatten and those muscles tighten as well. This is the opposite of how you play piano!
(Let those hands relax again).
Most new or returning adult students I see don't have much of an issue with getting a good hand curvature, but relaxing muscles while they play is much harder. Children tend to want to stretch their fingers out because their hands are small and they think, much like when using objects like doorknobs and forks and tennis balls designed for grown-ups' hands, they have to work at it really hard. But they don't need to do that, and as they progress and get the curve down, it's important to stress the hand and muscle relaxation that my adults sometimes struggle with.
Playing with a relaxed hand begins in the arms and wrists, which is another article. Don't hesitate to comment or ask me a question before then, and thanks for reading!