Place your fingers on the piano keys in your usual manner. Observe the position of your fingers. A common mistake that inhibits performance progress is improper finger shape. Do not lay fingers flat, because this will inhibit both speed and dynamics. Instead, form them into a pronounced arch, almost an upside down u-shape. Only the very tips of your fingers should be touching the keys. If you are not used to playing with this curved finger shape, it will seem awkward at first, but it will improve your skill level dramatically with practice.
Play a simple scale on the piano, paying attention to your finger placement. There are fingering patterns that can maximize playing ability, according to which scale is being played. Find the fingering patterns in your instruction book or online. Each finger is assigned a number from one through five. A number will be present above each note of the scale. One indicates the thumb, two the index, three is the middle finger, four the ring finger and five relates to the pinkie. Practice playing a scale repeatedly, using the correct fingering patterns, until it becomes natural for you. Do this with all the common scales.
Improve playing speed by using a piano instruction book specifically formatted for finger exercises. Start with the first exercises and practice until you can play it through with minimum mistakes and up to tempo. Move on to the second exercise and do the same. Once you have mastered the second lesson, go back to the first and continue into the second without stopping. Repeat this pattern with the rest of the exercises in the book, periodically reviewing past exercises to keep your skill level at its best.
Play a song from your piano book, paying attention to the numbers above some of the notes. As with the scales, the numbers indicate which finger to use to play that note. Play the song slowly, using the indicated fingering. Accelerate your speed until you can play the song up to tempo flawlessly, using correct fingering.