I want to express that after a certain experience I have some new insights. Does an insight come "into mind", or do you "gain/develop" insight? How is this best phrased?
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Gain insight is a very common (and rather generic) phrasing. Believe me, I know that, I tend to overuse it in academic writing ;-)
Other possible constructions include: “this experience provided me with insight into …”, “I developed a better insight into …”, “this intership offered plenty of insight into …”, etc.