Airplane [by-で] go [nominalizer-の] [subject-が] trip [purpose-に] go [purpose-のに] most fast.
Two uses of の: 飛行機で行くの (nominalizer for 'the act of going by airplane') and 旅行に行くのに ('for the purpose of going on a trip'). 一番 means 'most/number one' and forms the superlative.