Go west from Olivine City and use a Surfing pokémon to go into the water from the beach. You have quite a few trainer battles up ahead. Keep going in a generally southwest direction. You will see some whirlpools but you will not be able to go through them at this point.
Once you reach Cianwood, go to the Pharmacy to get some SecretPotion from the person there. This is what you need to cure Amphy.
You can go straight back to Olivine if you wish, but you will have an easier time if you defeat the Cianwood Gym leader first.
But before that, go to the northern area of town and you will see the elusive Suicune! It will run away, but a mysterious person named Eusine will appear and battle you to show Suicune that he is awesome. But you will defeat him!
After that's done with, go to the Cianwood Gym. The leader is Chuck. He uses fighting types, so don't bring rock or steel or ice or normal pokémon. Use flying and psychic moves on them.
Once you have defeated Chuck, you can use Fly outside of battle. You will get the Fly HM as you leave the Gym. Now you can warp to just about any pokémon center in all of Johto! Later on, you will discover that some places are too far to fly to. But this is not a problem, because you can just fly to a place close to where you want to go, and then you will be able to reach it after going closer to it.
Anyway, you can go west to check out the Safari Zone if you want, although it's closed right now. But otherwise, you should go back to Olivine with your SecretPotion.