You need the soul badge that is obtained from defeating the Fuschia city gym leader Koga in order to use surf outside of battle.
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The fifth badge is in Sootopolis City, you need Surf, Dive, and Whirlpool to get it
the one you get from the gym in fuschia city.
In Pokemon FireRed, the badge needed to surf is the Cascade Badge. This badge is obtained by defeating Misty, the Cerulean City Gym Leader. Once you have the Cascade Badge, you can use the move Surf outside of battle to navigate bodies of water and access new areas in the game.
it's the badge from crasher wake
Hearthome's badge