You can learn to play a piano song taking one of many possible paths. For a complete novice, the first song on the piano is usually learned "by ear", sounding out a familiar tune on the keys. If you are seriously considering learning to play the piano, it's really a great idea to go ahead an learn a bunch of tunes by ear, or by rote. Eventually, you may learn to read music, and develop higher level piano-playing skills. Then, you will be much less limited in what kind of piano music you are capable of learning and playing.
At this point, let's assume you are a beginner and you want to become a piano player. Here you will find a list of simple piano songs to get you started on the piano. Each one includes a picture of your starting position, the notes written in piano tablature (for those who do not yet read music), music notation (for those who want to see what it looks like, or who do read music), and a video demonstration.
These piano songs all can be played in multiple keys, but are offered here in the keys of C-major, F-major, and G-major (with the exception songs in minor keys, in which case you get A-minor, D-minor, and E-minor). Learning to play a single song in several keys is an outstanding way to develop your keyboard awareness, as well as your knowledge of the note names.
Follow the fingerings exactly as written on the music/tablature and as demonstrated in the videos. This will help you become comfortable and automatic with piano fingerings. We piano players have to know the piano-finger-numbers without even thinking. You can get there simply by learning many of these basic melodies with careful and focused attention on the fingerings. Don't let your fingerings be random.
You will find a thorough discussion of just how learning tunes on the piano benefits you on your piano journey in Chapter 4 of Piano Player... You, which is the ebook in which I tried to lay out the path for a successful building of your foundation of piano playing skills.
In the book, you will also find chapters on how to practice, the basic elements, scales & chords, music reading, and learning and memorizing music. For your interest in piano songs, I've included links here to some pieces that fall into the category of "learning and memorizing" music. Keep returning to this website in the future, as the number of piano songs offered will continue to grow.
Mary Had A Little Lamb in C, F, & G
Twinkle, Twinkle Little Star in C, F, & G
When The Saints Go Marching In in C,F, & G
You Are My Sunshine in C,F, & G
Down In The Valley in C,F, & G
Frere Jacques in C,F, & G
Alouette in C,F, & G
Yankee Doodle in C,F, & G
The Water Is Wide in C,F, & G
Brahms Lullaby in C,F, & G
Home On The Range in C,F, & G
Danny Boy in C,F, & G
Amazing Grace in C,F, & G
Simple Gifts in C,F, & G
America the Beautiful in C,F, & G
The Star Spangled Banner in C,F, & G
I have found this to be a great tune for use as a part of the following exercise routine: Scales, Arpgeggios, Cadences, and Burlesque
The video here provides a prime example of how to learn and memorize a piece.
Here are a couple of popular intermediate pieces, with information on how to learn them.