The badges you collect from defeating gym leaders will help you gain access to the Victory Road and Indigo Plateau after the last gym leader. The badges also help with controlling your Pokemon at different levels. Some badges also let you use HM moves, like fly and surf, outside of battle.
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rock smash isn't in Pokemon red version
get all the badges
yes
You can't find a Vulpix in Pokemon FireRed version. That pokemon is version-exclusive to Pokemon LeafGreen.