All four possibilities are OK: "What kind of noodle would you like?" "What kinds of noodle would you like?" "What kind of noodles would you like?" "What kinds of noodles would you like?" "Kinds" suggests the possibility that there be more than one answer. "Which" can be used in place of "what" but they are equally good. Use "which" if you want to emphasize that a selection is to be made.

kind = one type kinds = various types

Hey Heidi! I would suggest “What kind of noodles would you like?” and “Which kinds of noodles do you like?” The singular ‘kind’ of noodles refers to the decision that someone would make (they choose one kind). I suggest using ‘which’ with ‘kinds’ if your question will have more than one answer.