Locate sheet music for the songs you are interested in leaning to play. This can be done in several ways. One of the most cost effective ways is to visit your local library and search their musical scores. You may also wish to purchase a book though a music store that specializes in selling scores and piano/vocal songbooks, or directly from a publisher, which is easily done online.
Read the directions in the music, such as time and key signatures, key changes, repeats and tempi markings. Tempi is the plural of 'tempo,' or the speed at which the music should be played, which can vary throughout the composition. These directions are the "road map" to help you play the music accurately.
Practice the treble clef line, commonly for the right hand, several times on its own. Pay attention to any rhythms or finger patterns that are especially difficult for you, and allow yourself to practice those passages several times until you feel comfortable with them. Repeat this process separately for the left hand, or bass line.
Play both parts, the right and left hand (melody and accompaniment lines) together. Try this slowly at first for the sake of accuracy, then gradually speed up to the correct tempo.
Hum the melody as you play. This will help you get used to this added layer of music. Once you can confidently play the piano while you hum, try singing the actual words.