on one's best behaviour

be on one's best behaviour (= to behave very well in order to please or impress sb) — стараться вести себя как можно лучше;стараться зарекомендовать себя, показать себя с лучшей стороны (во время стажировки, испытательного срока);
|| примерно вести себя; вести себя как можно лучше

Example: Please let me come – I promise to be on my best behaviour. (MM)

Example: Gaga met President Barack Obama a few weeks back at a California fundraiser, where no, she didn't do something ridiculous or try to ambush the Pres, but instead acted modestly and appropriately. <...>
But even on her best behaviour, the singer managed to make an impression…or perhaps, her shoes did. As per usual, Gaga was rocking a ridiculous pair of heels, and President Obama admitted after event that he was a little taken aback by the singer's choice of footwear. "She was wearing 16-inc heels. She was 8 feet tall. It was a little intimidating," he said.

get/put sb on their best behaviour — наставить на путь истинный

Example: Get Kids On Their Best Behaviour (Reader's Digest)
Example: Putting Football on its Best Behaviour (Report by Dan Pope)

Related vocabulary:

straighten out
[браться за ум]
[clean up one's act]
[act your age]
[come to one's senses]
[settle down]