>>14167851The Pokemon League is located at a junction south of the map (marked in yellow).
1 is the windmill town, 2 is the crystal town, 3 is the ice town. It seems logical that you will go east of Lumiose City and come around in a semi-circle to these three towns, ending your journey at the gates of the League.
1 is likely Flying-type - the badge might be a wing of a bird or a plane. 2 is likely Fairy-type to tie in with the crystal monument. 3 is likely Ice-type and the snowflake-like badge agrees with this.
