Yes, it can! Check the table on the Flying Press move page, which shows how effective the move is against each type. It is super effective against Normal, Grass, Ice, Fighting and Dark.
That means it will be 4x effective against any combination of those such as Fighting/Dark or Grass/Ice. FYI, Gogoat and Skiddo are pure Grass so Flying Press will only be 2x effective.
Also note that since there is no crossover of the things Flying/Fighting are super effective against, it can never be more than 4x effective.