[noun] + が + 下手
[verb (dictionary form)] + の + が + 下手 [verb (dictionary form)] + こと + が + 下手
下手 is about skill: the result is poor, whoever judges it. にがて苦手 is about you: you are weak at it, or would rather avoid it. Something that is not a skill – getting up early, a food, a kind of person – only takes 苦手.
Calling your own skill 下手 is ordinary modesty, and it is the usual reply when someone praises you. Said about someone else, 下手 is blunt; あまりじょうず上手じゃないis gentler.
Before a noun, 下手 is a な-adjective and takes な.