If you read any biography of his life you'll find that Frank got his first big break thanks to that time honored show business trope, the pushy stage mother. In his case his mother, Dolly, suggested that maybe the Three Flashes needed another voice and with Frank they appeared on the Major Bowes Amateur Hour, the American Idol of its day, as The Hoboken Four in 1935. Unsurprisingly, they won.
After he left The Hoboken Four he performed as a singing waiter and recorded at least one demo with Frank Mane's band. In 1939 he was signed by Harry James to sing with his orchestra and his first released recording was this one, "From the Bottom of My Heart."
While Harry James was a gracious boss, releasing Frank from his contract when the next gig came up later that year, his next boss was...well, I wish Frank had a lawyer or agent on his side, put it that way. Tommy Dorsey's was one of the biggest of the big bands and when he hired Frank away from Harry James his contract stipulated that he was entitled to "one-third of Sinatra's lifetime earnings in the entertainment industry [thank you, wikipedia]." The way that's worded that means that Frank owed Dorsey one-third of his paycheck from everything he ever did, including his Oscar-winning role in From Here to Eternity and any appearances with the Rat Pack. Are you hearing strains of "Speak Softly Love" in the distance? You totally should be because it's this contract with Tommy Dorsey (and the rumor of Sam Giancana's involvement with Frank's release from it) on which the relationship between Don Corleone and Johnny Fontane in The Godfather is based. The truth of the matter is here. In any case, Frank recorded some classics with the Dorsey band including this one, "I'll Be Seeing You."
It's kind of up in the air who was the original teen idol sensation, Sinatra or Rudy Vallee from the 20s (Vallee was famously mobbed by flappers). There are old news reels of legions of bobby soxers waiting in lines for hours to get into Sinatra's matinee shows, kissing his picture and screaming. You thought that all started with Elvis? Nope, it went back at least ten years to Frank. According to the Astro Turf entry over at tvtropes.org at least some of those hysterical bobby soxers were actresses paid by Sinatra's publicist (scroll down the page and it's the first example under Real Life: Music). In 1943 Frank signed with Columbia Records but was unable to release any new recordings due to a two year strike led by the American Federation of Musicians so instead they rereleased "All or Nothing at All," recorded in 1939 with Harry James.
Much later in his career Sinatra worked with another bandleader from New Jersey and we'll be seeing him on our next stop. Have a great day whatever you're doing and thanks for your patience.