Learn Hardcore Korean: babeul meokgo naseo keopireul masyeoyo.

-고 simply links two verbs: “eat and drink.” It can imply sequence, but not explicitly that the first action finishes before the second starts.
-고 나서 literally means “after doing (verb).” It emphasizes that the eating is fully completed before the coffee‐drinking begins. In English it corresponds to “after (I) eat, (I) drink coffee.”

Why doesn’t 먹고 나서 have a polite ending like -요? Why is -요 only on 마셔요?

Korean politeness markers (like -요) attach only to the final verb or adjective in a clause. Connective forms such as -고 or -고 나서 are inside the sentence, so they don’t carry -요 themselves. The sentence ends with 마셔요, which makes the whole sentence polite.

Why do we say 밥을 and 커피를? Can we omit -를?

을/를 marks the direct object of a verb (here, 밥 for 먹다, 커피 for 마시다).
In casual speech or notes, Koreans often drop object particles: 밥 먹고 커피 마셔요 is perfectly natural in conversation. Including -을/를 makes the sentence more precise and is standard in written or formal contexts.

What does 밥 mean here? Does it literally mean “rice”?

While 밥 literally means “cooked rice,” in everyday Korean it often means “meal” in general. So 밥을 먹다 is “to eat a meal,” not necessarily only rice.

Why is the subject omitted? Who is doing these actions?

Korean frequently drops subjects when they’re understood from context. Here, the implied subject is “I” (or “we,” depending on context). It’s clear from context who’s eating and drinking, so there’s no need to state 저는 or 우리는 explicitly.

Can we use 먹은 후에 instead of 먹고 나서? What’s the nuance difference?

Yes. 먹은 후에 = “after eating.” It’s formed by turning 먹다 into the noun 먹은 + 후(에) (“after”).

먹은 후에 tends to sound a bit more formal or written.
먹고 나서 is more conversational, emphasizing “once you’ve finished eating.”

Why is 마셔요 in the present tense if this sentence describes a sequence? Wouldn’t past tense be needed?

This sentence uses present tense to describe a habitual or general sequence: “I eat (my meal) and then I drink coffee.” If you want to narrate a past event, you’d switch to past tense: 밥을 먹고 나서 커피를 마셨어요 (“After eating, I drank coffee.”).

Could the word order change? For example, can I say 커피를 마시고 밥을 먹어요?

You can swap clauses grammatically, but the meaning changes to “I drink coffee and then eat.” If you intend “drink coffee after eating,” you must put 밥을 먹고 나서 first. In Korean, sequence connectors usually appear in the order things happen.