take the cloth - Idioms - by the Free Dictionary, Thesaurus and Encyclopedia.) , but I don't think there's an idiom for becoming a monk. A monk just joins an order (although you could use 'the' if the context made it clear that there was only one possible order - if, for instance, all monks in his home-town were Franciscans [for some reason ]).
A monk and a priest also 'take vows'; take care with this idiom - 'take the vow' can have various other meanings (marriage, abstinence from sex/alcohol... )
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