John Lennon was a young rock n roll enthusiast in the 50s and started with the banjo, which her mother taught him to play. He soon picked up an acoustic guitar from the local store and started playing around with it. With this new found medium to express his love for rock n roll he formed a band called the Quarrymen. 

One day the Quarrymen were playing a small show at a church fete, Ivan Vaughn (a common friend of Paul and John) invited Paul McCartney to watch Lennon perform with his band. Post their performance Ivan introduced the two and they soon realized their shared love of rock and roll when McCartney played Twenty Flight rock on the guitar for him, John loved it and invited Paul to join the band.

Paul later introduced his school pal George Harrison to John as a potential guitarist for the Quarrymen, but John turned him down saying George(15 years of age at the time) was too young. Later Paul had George play Raunchy for John in a bus, impressed by his skills John took George in.

In 1960 they changed their name from the Quarrymen to the Beatles. Later that year while playing a gig in Hamburg, their drummer Pete Best wasn't available, that's when they met Ringo for the first time, he was playing with Rory and the Hurricanes at the time and filled in for Pete Best for the Beatles.

They were amazed by what he did on the drum set and knew instantly that he was the one. He just played and it fit.

In August of 1962 they invited Ringo to join them and fired their then drummer Pete Best. The rest is history.