들어가다 literally means "to enter" here. Hence a literal translation would be: "Son's friends who are entering the stage of self-introduction" "자기소개 하는" would technically mean the same, but as you might have noticed already, "to enter the stage of something" sounds overblown compared to "to start something", in addition to being indicative for it being part of a (lengthy) show. Here it becomes clear that these kids are (most probably) barely more than toddlers performing a show taught by their parents/teachers, and the adults are watching it with great pleasure and love.

들어가다 here means "to start, to begin" It's closer to "또 자기소개(를) 시작하는 아들 친구들" than to "또 자기소개(를) 하는 아들 친구들" You can also say 자기소개에 들어가는, but 자기소개에 시작하는, 자기소개에 하는 are ungrammatical. 자기소개를 들어가는 is ungrammatical in general, but can be accepted by more causal, colloquial speech

Thank you for your comment Jake. ^^ I've had a few great teachers along the way, but have mostly been self-studying. I've been trying to take a more natural approach. That is, I listen to normal Korean conversations (whether it is through a video or in person) and try to imitate or copy the phrases and expressions.

Your Korean is truly amazing. You must have a great tutor, and/or you simply know how to learn a foreign language properly. Either way, any native speaker would recognize the effort you have put in learning the language. Kudos.

dd