Getting to Olivine City

Go west from Ecruteak. You will fight some trainers along the way, and there is an Apricorn Tree here. You will pass by Miltank Farm, where a sick Miltank needs 7 Oran Berries to get well. If you don't have enough, you can try to enter the random drawing at Goldenrod Department Store over and over. Sometimes the third prize is an Oran Berry.

Once Miltank is well again, the owners of the farm will sell you Moomoo Milk, which recovers 100HP. You can buy one or a dozen at a time.

Going south, you will reach Olivine City. The Gym Leader isn't there because she's tending to the Ampharos in the lighthouse, who is sick.

Go to the lighthouse and fight your way up. You won't be able to reach the Gym Leader by way of the Elevator, so you will have to go the long way. There's a part where you will have to go out through a broken wall, so look around if you are stuck. Eventually you will reach the Ampharos and Jasmine, who humbly requests that you go to Cianwood City to get medicine for Amphy. To do that, you will have to go west.