Learn Russian: Он никогда не пробовал наркотики.

When talking about general life experiences (whether someone has ever done something or never done it), Russian strongly prefers the imperfective aspect. Since the sentence uses никогда не (never), we use the imperfective пробовал to state that the event was entirely absent from his life.

Could I use the genitive case наркотиков here because of the negative не?

Yes, you could. The genitive of negation (не пробовал наркотиков) is grammatically correct and emphasizes the complete absence of the object. However, in modern everyday Russian, using the accusative case (наркотики) after a negated verb is very common and completely natural, especially for concrete or countable nouns.

Is there a difference between пробовать and пытаться when translating the English word 'try'?

Yes. Пробовать means to 'try' in the sense of testing, sampling, or experiencing something (like food, a new activity, or drugs). Пытаться means to 'attempt' or make an effort to do an action, and is typically followed by another verb (like trying to open a door).

Does наркотики only refer to illegal drugs, or can it mean medicine too?

In modern everyday Russian, наркотики almost exclusively refers to illegal recreational drugs or narcotics. If you want to talk about medicinal drugs or prescriptions, you should use лекарства (medicines) or препараты (medications).