"I got to know a new word" sounds strange upon first hearing it, and I think its very likely people will take the meaning wrong. I may even be taking the meaning wrong. But "got/get to know" is a common English expression. I don't see any reason why you can't "get to know" a word. If that was your intended meaning, I would say it suggests a deeper knowledge of the word than just learning its definition from the dictionary. Typically, however, we "get to know" people, or places, or otherwise things of a rather complex nature.