It is an HM that can be used to traverse any body of water, usually very needed in the game to access areas that one could not access without it.
At some points in the game, if you can't surf, you will not be able to advance any further.
It is also an extremely cool water attack, but in double battles, it can hurt or even Faint out a teammate.
I think that it might usually be learned around the 4th Gym Battle, but it really depends on which version you are playing.
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No. Seviper cannot learn surf.
No, Togepi cannot learn Surf.
Vanillite cannot learn Surf.
Slowpoke does not learn Surf by levelling up, but it can learn Surf via HM in all generations.
Yes, Zigzagoon is able to learn Surf.