The distance from "Maybe I can learn to play piano ..." to playing pieces from the masters such as Mozart, is not as far as you might think! Many famous composers have simpler pieces, sections, or movements in their publishing credits, some of which were written to help their students or children develop and hone particular skills, leaving us with a treasure chest of repertoire for every level of piano.
When considering which repertoire books to use, choose curriculum
that incorporates a variety of songs and styles from the Baroque, Classical, and Romantic periods, as well as more modern pieces. Not only will this expose you or your child to various styles and techniques, but to musical beauty and genius. Remember, some of these pieces, though hundreds of years old, are enjoyed around the world to this day by musicians and audiences alike.
As an RCM Certified teacher, I am a little partial to the Royal Conservatory's Celebration Series, as it includes this criteria at virtually every level in a systematic and progressive manner of study, consistently building strong musicianship skills. Here you may find, for example, in the elementary levels, delightful pieces that Mozart wrote as a child or that composers such as Bach or Czerny wrote for children or students.
You don't have to be a virtuoso or a well-advanced student to enjoy playing some of the best music ever written! Good repertoire, a good instrument and a good teacher will get you on your way ... beautifully.