The first song Sinatra ever recorded was
Our Love, a demo recording with conductor/ arranger Frank Mane cut on 18 March 1939 in New York City. The first recordings that got released were
Stardust and
Wishing Will Make It o, recorded at the Roseland Ballroom, New York on 8 July of the same yearafter Frank had joined Harry James and His Orchestra. But don't take my word for it - check this phenomenal Sinatra 'sessionography':
http://www.jazzdiscography.com/Artists/Sinatra/ind
ex.html