Hello ledojaeger,
日本語で簡潔な説明を示してください = Not exactly wrong (actually, I would say quite good for a beginner), but I find the original sentence Please demonstrate a concise explanation in Japanese to be kind of a mouthful. Wouldn't we normally just say "give me an example" or perhaps "could you explain that for me". As it is I don't know if you want a demonstration or an explanation. Japanese tends to be an economical language: if it can be said simply, it should be said simply. So, could you just say; "please explain it simply in Japanaese"?
わかりやすく説明してください or わかりやすく説明してくれますか if you want to sound like even more like you are asking for a favor.
単語 use for individual words. Probably going to be your best choice for almost all situations. 語句 I think refers to a phrase/sentence comprised of a number of words.
I want to get to know you or I want to know you better = あなたのことをもっと知りたいです.
Although, as in English, it is hard to say this without imparting at least a slight nuance of "I'm interested in you\".