I checked on internet to find the related meaning of "系" in below sentence but I didn't find. "レポート 進んでる系?"
This is slang.
Originally, 系 is a technical-sounding suffix that means "family", "(sub)system" or "series" (see this). For example, 太陽系 means "solar system". 系 is also used to categorize genres/types of art, people etc (eg ジャズ系の音楽, 北欧系の人, 小動物系の女性).
In slangy speech, people use 系 somewhat freely and say "アニメが好き系？" ("Are you the anime-loving type?"), "よく食べる系の人" ("type of person who eats a lot"), and so on. Some young people even stretch this and use 系 almost meaninglessly, and "レポート進んでる系?" is such an example. Its literal meaning is something like "Are you the type whose report is in progress?", but you can translate it simply as "What's up with your report?". Likewise, if a charai person says "一緒にお昼食べる系?", it just means "How about having lunch with me?".