Badges have passive abilities. Just having it activates any abilities it might give.
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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.
Cerulean city
you cant trust me
A tree will be blocking the way so you will need to use Cut on the tree to get past it. You can get Cut from the Captain of the SS Anne. You need the 2nd badge (Cascade Badge) in order to use Cut outside of battle.
The 8 badges for Pokemon Red are: Boulder Badge, (Brock) Cascade Badge, (Misty) Thunder Badge, (Lt. Surge) Rainbow Badge, (Erika) Marsh Badge, (Sabrina) Soul Badge, (Koga) Volcano Badge, (Blaine) and Earth Badge. (Giovanni)