No, Togepi cannot learn Surf.
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First you have to go the Five Island, then surf around and look for a man with a beard and brown clothes, then talk to him and you'll get an egg which will hatch into Togepi
No. Seviper 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.