Learn Hardcore Korean: jumare chingudeulgwa noraebangeseo noraereul bulleoyo. - On weekends I sing at karaoke with friends.

-에 on a time word marks “on/at/in (time).” So 주말에 = “on the weekend.”
-에서 is for the place where an action happens, not for time, so you can’t use it with 주말.
For contrast/emphasis on the time, you can say 주말에는 (“as for the weekend, …”).

Why is there no subject? Who is doing the action?

Korean often drops the subject when it’s obvious. In everyday conversation, it usually means “I/we.” You can add it if you want:

What exactly does 친구들과 mean? Do I need 들 to make “friends” plural? Could I say 친구와?

친구들 = “friends” (plural marker -들). It’s optional in many contexts, but it makes the plurality explicit.
-과/와 can mean “and/with,” so 친구들과 = “with friends.”
친구와 usually reads as “with a friend” (singular), though Korean sometimes leaves number to context.
In casual speech, even 친구랑 can imply multiple friends if the context makes it clear.

What’s the difference between 과/와, 하고, and (이)랑 here?