The following is how Soncino (correctly) translates the relevant portion of the story:
'Are you as the son of Lakisha?' he complained: 'when I stated a law, the son of Lakisha used to raise twenty-four objections, to which I gave twenty-four answers, which consequently led to a fuller comprehension of the law; whilst you say, "A Baraitha has been taught which supports you:" do I not know myself that my dicta are right?'
While the Aramaic word is קושייתא, which means questions, the true meaning here is not inquisitiveness. Resh Lakish was not trying to find out more details or to understand the depth of the reasoning of the law.
Rather, Resh Lakish had been arguing with Rabbi Yochanan. The 24 kushyasa were 24 objections. Resh Lakish had mastered the Tannaitic material, and when Rabbi Yochanan stated a law, Resh Lakish argued that this law did not accord with this source or that that source. In response, Rabbi Yochanan had to explain how each of these sources was different from his case. In contrast, Rabbi Eleazar ben Pedat found a support rather than a contradiction in the Tannaitic material. Rabbi Yochanan yearned for someone to argue with.
While inquisitiveness can indeed be a positive trait, this is not necessarily what I would derive from this story.
To become like Resh Lakish and be able to come up with these objections, one would first need to memorize and review Tannaitic literature (or whatever the source material would be in your Torah topic of choice), develop a comfort and mastery over it, and be able to call up that material in order to challenge any newly introduced idea.