well, those two are used in totally different way.

~기 is like ~ing.

I mean, ~기 is used when you change a verb to a noun.

ex)밥 먹다(eat a meal) ->밥 먹기 (eating a meal)

    말하다(speak,talk,chat...)->말하기(speaking, talking, chatting)

~고 싶어요 is used when you say your wish.

It means "I wish/I want/I hope/I'd like to"

~고 싶어요 is combined with verb.

ex)영화 "보고" 싶어요(I want to "see" a movie)

    옷을 "사고" 싶어요(I'd like to "buy" clothes.