Where to Put your fingers
For scales:
The orthodox technique, for example if you are going up the scale with the right hand, would be to play the first three notes starting with your thumb, and then after playing the note under your middle finger to play the next note with your thumb again. (The thumb crosses under.) Going back down the scale, starting with the pinky on your right hand, you would play five notes, and then your middle finger would cross over the top of the thumb to play the next lower note.
Playing with the left hand is a mirror image of what the right hand does.
For regular music:
The same sort of arrangement applies, but it is also common to just pick up your hands and move them so that they span the next sequence of notes you are expecting to play.
A Parting Word
And finally, relax. It's so easy with quick moving songs to let the fingers get all tensed up with the challenge of being on the right key at the right time.