https://www.woot.com/blog/post/the-debunker-did-the-monkees-play-their-own-instruments

Quote from Ken Jennings, known for being knowledgeable about a wide range of topics:

The common rap on the Monkees, then as now, is that they were TV fakes who didn’t even play their own instruments. It’s true that, originally, the Monkees’ music was supervised by Don Kirshner (later of The Archies fame!), who didn’t want even Nesmith and Tork, both gifted musicians, playing on the recordings. But Nesmith was allowed to write and produce a few tracks on each of the first two Monkee LPs, and he brought Tork in to play alongside the session musicians, even if he himself had been banned from playing guitar on his recordings. The same year, 1966, the Monkees began touring, so Mickey Dolenz quickly picked up the drums, while Davy Jones played tambourine and eventually got proficient enough on rhythm guitar, bass, and drums to fill in when necessary.

More in depth:

https://medium.com/cuepoint/fake-it-til-you-make-it-how-the-monkees-performed-live-f9fea6c9a6b9

“I was standing at a place we were playing. We were backstage and it’s like two minutes before we’re supposed to go on. And this guy walks up to me, he’s a reporter you know, like that anyway. I’m standing with my guitar over my back, he walks up to me and says, ‘Is it true that you don’t play your own instruments?’ I said, ‘Wait a minute! I’m fixin’ to walk out there in front of 15,000 people, man. If I don’t play my own instruments I’m in a lot of trouble!’”— Michael Nesmith, January 1967

For three months we practised our music. When you don’t know a thing about music it’s a little hard to keep the beat. I had never even picked up an instrument, but Mike, Micky, and Peter were great on guitar. We just played for something to do, and Screen Gems rented the instruments for us. We decided someone would have to play the drums and Micky volunteered, though he couldn’t really play them — he couldn’t keep rhythm. Peter got to be the bass guitarist because Mike didn’t want to play it. — Davy Jones

The first public appearance the group made was, it may surprise the reader to learn, playing live.

https://en.wikipedia.org/wiki/Peter_Tork

Tork began studying piano at the age of nine, showing an aptitude for music by learning to play several different instruments, including the banjo, acoustic bass, and guitar. … He attended Carleton College before he moved to New York City, where he became part of the folk music scene in Greenwich Village during the first half of the 1960s.

Tork was a proficient musician before he joined the Monkees. Though other members of the band were not allowed to play their instruments on their first two albums, he played what he described as “third-chair guitar” on Michael Nesmith’s song “Papa Gene’s Blues” on their first album. He subsequently played keyboard, bass guitar, banjo, harpsichord, and other instruments on the band’s recordings. He co-wrote, along with Joey Richards, the closing theme song of the second season of The Monkees, “For Pete’s Sake”.

The DVD release of the first season of the show contains commentary from various band members. In it, Nesmith states that Tork was better at playing guitar than bass. Tork commented that Davy Jones was a good drummer, and had the live performance lineups been based solely on playing ability, it should have been him on guitar, Nesmith on bass, and Jones on drums, with Micky Dolenz taking the fronting role (instead of Nesmith on guitar, Tork on bass, and Dolenz on drums). Jones filled in briefly for Tork on bass when he played keyboard.

https://en.wikipedia.org/wiki/Michael_Nesmith

After Nesmith’s tour of duty in the Air Force, his mother and stepfather gave him a guitar for Christmas. Learning as he went, he played solo and in a series of working bands, performing folk, country, and occasionally rock and roll. He enrolled in San Antonio College, where he met John London and began a musical collaboration. They won the first San Antonio College talent award, performing a mixture of standard folk songs and a few of Nesmith’s original songs. Nesmith began to write more songs and poetry, then moved to Los Angeles and began singing in folk clubs around the city. He served as the “Hootmaster” for the Monday night hootenanny at The Troubadour, a West Hollywood nightclub that featured new artists.[9]

Randy Sparks from the New Christy Minstrelsoffered Nesmith a publishing deal for his songs.[8] Nesmith began his recording career in 1963 by releasing a single on the Highness label. He followed this in 1965 with a one-off single released on Edan Records followed by two more recorded singles; one was titled “The New Recruit” under the name “Michael Blessing”, released on Colpix Records—coincidentally this was also the label of Davy Jones, though the two men did not meet until the Monkees were formed.[10]

Once he was cast, Screen Gems bought his songs so they could be used in the show. Many of the songs Nesmith wrote for the Monkees, such as “The Girl I Knew Somewhere”, “Mary, Mary”,[8] and “Listen to the Band” became minor hits.

https://en.wikipedia.org/wiki/Micky_Dolenz

Dolenz originally had his own rock band called “Micky and the One-Nighters” in the early- to mid-1960s with himself as lead singer.[5] He had already begun writing his own songs. According to Dolenz, his band’s live stage act included rock songs, cover songs, and even some R&B. One of his favorite songs to sing was Chuck Berry’s “Johnny B. Goode”, which he sang at his Monkees audition, resulting in his being hired as one of the cast/band members.[citation needed] He recorded two 45s in 1965 that went unreleased until the Monkees’ success in 1967.