My baby book does not pick out any one song as first, but it says (in my mother's hand, ca. 1940),

By 2 1/2 years, knew about 25 nursery rhymes, same most of them, carrying tunes perfectly.

_Hymns_
Lead Kindly Light
Jesus Bids Us Shine
Blessed Jesus
Silent Night

Frere Jacques
Au Claire de la Lune
Aupres de ma Blonde

Kommt a Vogel
Muss i denn
Ah wie ist's moeglich
Von meinem Bergli
Guten Abend Gute Nacht

Oifn Pripitchok
Rozhenkes mit Mandeln

Come, Let's Play We're Indians
Here Comes the Sandman
Lovely May

The first song I ever *heard* was almost surely Rozhenkes mit Mandeln, mentioned above. I have the text, in my mother's hand, that she learned it from while in the hospital to have me.

On my own, I remember that my favorite tune on the jukebox, when I was about 3, was St Louis Blues. It was No. 4 and cost a nickel.