What was The Beatles’ first gig?

There are actually a number of candidates for this answer.

1. The Quarrymen’s first gig. I haven’t been able to pin this down exactly, but the first live performance of The Quarrymen (aka The Quarry Men) seems to have been in May 1957, because the generally reliable site The Quarrymen enter the TV Star Search competition notes that they entered a TV talent show competition on 9 June 1957, ‘less than a month after their first live appearance’. Less than a month later would come their performance at Woolton Fete on 6 July 1957, the first verified occasion that John Lennon met Paul McCartney.
2. The Quarrymen’s first gig with Paul McCartney. This was on 18 October 1957 at the New Clubmoor Hall at Norris Green, Liverpool. McCartney had been invited to join the band back in July, but he’d deferred doing so on account of his Scout camp and summer holidays with his family.
3. The Quarrymen’s first gig with George Harrison. Harrison joined the band in March 1958 but I haven’t been able to tie down when he first appeared on stage with them. He did, however, join them for the recording session at Phillips’ Sound Recording Services on 12 July 1958.
4. The Quarrymen’s first time sharing a bill with Ringo Starr. This was on 13 March 1959, at the Morgue Skiffle Cellar in Liverpool, with the headline band being Al Caldwell’s Texans, featuring Starr on drums. Al Caldwell would later adopt the stage name Rory Storm, and his band became the Hurricanes. The Quarrymen were rather lower down the bill.
5. First gig as ‘Johnny and the Moondogs’. This was on 18 October 1959, at a second audition for TV Star Search; it’s likely that they changed their name in the hope of not being recognised. They were no more successful on this occasion than on the previous one, although it may have been on this occasion that Lennon, whose cheap guitar had finally given out some time earlier, walked away with someone else’s.
6. First gig as the ‘Silver Beats’. Stuart Sutcliffe joined the band in January 1960 and he and Lennon decided in March 1960 to change the band’s name to the ‘Beatals’, but this didn’t seem quite right as they failed to settle on a definite spelling of it. They toyed with ‘the Silver Beetles’ before playing a gig at Lathom Hall in Liverpool as the ‘Silver Beats’ on 14 May 1960.
7. First gig as the ‘Silver Beetles’. This was a few days later at the Iron Door Club, on 15 May 1960. After this, the band would go on a brief Scottish tour backing singer Johnny Gentle, in which they were billed only as ‘Johnny Gentle and his band’.
8. First gig as the ‘Beatles’. As far as I can tell, this was their first night in Hamburg: 17 August 1960 at the Indra Club, the first of 48 such nights. This being a weeknight (Wednesday), they played for four hours: 8-9.30pm, 10-11pm, 11.30pm-12.30am and 1-2am.

(The Beatles at the Indra Club, 17 August, 1960.)

It only remains to give one more date:

• The Beatles’ first gig with Ringo Starr as a permanent member. This was on 18 August 1962 at the Hulme Hall in Port Sunlight in Merseyside.