Learn Hardcore Korean: sueobeul machigo doseogwane gayo. - I finish class and go to the library.

The ending -고 is a connective suffix that links two actions in sequence: “finish (A) and then go (B).” It’s a simple way to say “after doing A, do B.”

Could I say 수업이 끝나고 instead of 수업을 마치고?

Yes, you can. 끝나다 is intransitive, so it takes the subject particle 이/가 (e.g., 수업이 끝나고). The nuance is nearly the same:

수업을 마치고 (using 마치다, “to complete/finish something”)
수업이 끝나고 (using 끝나다, “something ends”)

Why is 도서관에 marked with 에 instead of 에서?

The particle 에 marks the destination or goal of movement when used with verbs like 가다 (“go”). In contrast, 에서 marks where an action takes place or the starting point. Here, you’re going to the library, so you use 도서관에.

Why is the verb 가요 used, and what speech level is it?

가요 is 가- (stem of “to go”) + -아요 (polite informal ending). This makes it present/future tense in polite speech (존댓말). It’s commonly used in everyday conversation when you want to be polite but not overly formal.

Where is the subject in this sentence?

Korean often omits subjects when they’re clear from context. Here, the implied subject is “I” (저/나), so (제가) 수업을 마치고 도서관에 가요 becomes simply 수업을 마치고 도서관에 가요.

What’s the difference between connecting with -고 and -아서/어서?

-고: Pure sequence (“A and then B”), neutral.
-아서/어서: Can indicate sequence but often carries a nuance of cause or reason (“A so B”). For a simple chronological order, -고 is more straightforward.

Could I use a more formal connector, like -고 나서 or -ㄴ 후에?

Yes. If you want to be slightly more formal or emphasize the order:

수업을 마치고 나서 도서관에 가요
수업을 마친 후에 도서관에 가요
Both mean “after finishing class, I go to the library,” with -고 나서 and -ㄴ 후에 adding a bit more explicit sequence.