"Do you like going to karaoke?"
I don't underatand why "iku no ga suki" uses two particles in one place. I don't think I've seen this before.
You can have multiple particles. Theの here is marking the verb a noun, and the が is marking our now noun/gerund "going" as the subject.
I think GA makes it an object in this case.
Object a person or thing to which a specified action or feeling is directed.