Pokémon Gold and Silver/Cianwood City

Get the Secretpotion
The man in the pharmacy at the bottom of Olivine City will give you a Secretpotion once you talk to him, which you need to bring to Jasmine in the lighthouse in Olivine City. It'll cost you nothing, and you won't be able to battle Jasmine until you get this and bring it to her, so this should be your first stop in Cianwood.

Get a Shuckle
In the house directly north of the pharmacy lives a boy who had a stolen from him, and he wants to give his other Shuckle to you for safekeeping! The only other way to find Shuckle is to use Rock Smash in the field, and hope to find one, but this way is much easier, so accept the boy's offer and care for that Shuckle!

Fly
This woman is the woman your gratitude to - she is Chuck, the Cianwood Gym Leader's wife, and she thinks her husband is spending too much time at the gym. Once you beat Chuck, talk to her and she gives you the item you will almost certainly enjoy more than any other (with the possible exception of Surf): HM02, Fly. HM02 allows you to visit any city you've previously been to, including Olivine, which saves you the entire trip back.

Rock Smash
If you have a Pokémon that can use Rock Smash, head to the north end of Cianwood and use Rock Smash on the breakable rocks. Under many of the rocks you'll find Krabby's, but near the north end one of the rocks contains a Revive.

Gym Battle 5: CHUCK
The trainers at the Cianwood City Gym are all fighters, and their Pokémon are of that same type. As such, Ghost Pokémon are completely immune to many of their attacks, but Flying and Psychic types are the Pokémon that really dominate here. A high-level Kadabra, Hypno, or a Pidgeotto with Wing Attack (which it learns at level 33) can do wonders here.

After defeating Chuck, you get the Storm Badge, which enables you to control Pokémon up to level 70 and use Fly outside of battle, and he also gives you TM01 which contains Dynamic Punch, a move that hits only 50% of the time, but confuses the enemy every time it hits.