Did I use the word ‘train’ correctly? Is the grammar correct?
There are many things you could say with slightly different meanings.
She showed him how to repair the bicycle. (A specific bicycle has a specific problem)
She showed him how to fix the bike. (Same, more naturally)
She trained him to repair bicycles.
She trained him how to repair a bicycle. (She gave him general training so that he could repair many kinds of problems on many kinds of bicycles. Training has the sense of being substantial.)