direct translation is 너와 이야기하게 되서 좋았어. but this is unnatural. natural expression would like belore 이야기 하게 되서 좋았어(반말) 대화 하게 되서 좋았어요(높임말) 대화 나누게 되서 좋았습니다.(극존칭어) omit "I(나, 저)" would sound more natural.

I'm glad to talk to you. would be translated as 너와 이야기하게 돼서 기뻐. Your attempt "너한테 이야기해서 기뻐" is grammatically correct. But it sounds unnatural. But, the grammatical correctness is not the whole element of good sentence. You need to consider native Korean way of thinking 1) 이야기하다 is not a one-sided action but a mutual action. So, 이야기하다 always used as "와/과(with)" to talk to A => A 와/과 이야기하다 2) 이야기하다 is not a moment action but a long duration action. The feeling 기쁘다/반갑다 usually applies to a moment or situation. So, 기쁘다/반갑다 is often used with "게 되서(glad of the situation of doing something)"