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.