Purchase a beginning piano method. Alfred, Shaum and Bastien are some of the most popular methods. Bastien features a well-rounded approach to learning piano that includes level-appropriate theory lessons as well; if you want to pursue music seriously, you might want to consider this method. Shaum introduces classic piano music at every level; this might be the best method if your main goal is to begin playing piano classics right away.
Practice your selected piano method, but don't wait until you have mastered the piano to begin playing piano classics. There will be simplified versions of some of these classics included in your piano books. When you feel comfortable playing those versions, purchase simple piano sheet music from a music retailer or visit one of the many online libraries that provide free simple piano classics (see Resources). Try your hand at classical, pop and Christmas classics for variety.
Continue working through the levels of your piano course while augmenting your lessons with simple piano classics such as "Fur Elise" or the popular "Chopsticks." As you advance through your books, add more difficult versions of the simple piano classics you have been playing. Sheet music for piano comes in various arrangements to fit the level of your playing, and by playing more complex versions of the same song as you progress in skills, you will find the transition easier. You also can add new classics to the mix to expand your musical repertoire.