Learn Hardcore Korean: chinguege yorireul baewoyo.

친구에게 breaks down to friend + the dative marker 에게. This particle marks the person from whom an action originates or to whom something is directed. With 배우다, it indicates the source of learning, so 친구에게 배우다 means learn from a friend.

Why is 요리 followed by 를?

The particle 를 marks the direct object of an action. Here 요리 means cooking, which is what is being learned, so 요리를 배우다 means learn cooking.

What does 배워요 indicate in terms of tense and speech level?

배워요 is the present-tense polite informal form of 배우다. It expresses a habitual or ongoing action in a polite but conversational style, equivalent to I learn or I am learning.

Why is there no subject in the sentence?

Korean often omits pronouns when the context is clear. Here the implied subject is I. A preceding conversation or situation usually makes it obvious, so the pronoun is dropped.

Can 에게 be replaced with 한테 here? Is there a difference?

Yes. 친구한테 요리를 배워요 is more colloquial. 에게 is neutral or slightly more formal, while 한테 is casual and very common in spoken Korean.

How would you express the same idea in the past tense?

Change 배워요 to its past form 배웠어요. The sentence becomes 친구에게 요리를 배웠어요, meaning I learned cooking from a friend.

If I want to say my friend taught me cooking, not that I learned, how would I phrase it?

Use the verb 가르치다 meaning teach, and mark the teacher as subject. For example 친구가 나에게 요리를 가르쳐요 means my friend teaches me cooking.

Can I rearrange the word order and still keep the meaning?

Yes. Korean word order is flexible because particles mark roles. For example 요리를 친구에게 배워요 or 요리를 배워요 친구에게 convey the same idea, though the last is less common. Typically the indirect object precedes the direct object.