Pokémon Red and Blue/Vermilion City

Do some Pokémon Trading
The trader won't deal with Yellow players, but they can talk to the girl just outside for a free once they've won the Thunder Badge. She is hoping that you will take good care of it.

Red/Blue players can trade a for a very rare  in the building above and to the right of the gym. Don't hesitate to make this great trade if you picked up the fairly common Spearow! If you don't have a Spearow to spare, you'll get a chance to grab one in the upcoming Route 11

Many important items
Convince the old man in the building left of the Pokémon Center that you love fishing as much as he does and he'll take a shine to you. He will give you his Old Rod as a gift! Now you can use it near any body of water and get... a. Or... another Magikarp. Yes, that's all this Rod catches. Good news if you didn't buy one earlier.

Something slightly cooler than the Magikarp Stick can be found at the Pokémon Fan Club, located directly above the gym. Listen to the president's boring story and he'll give you his Bike Voucher. Redeem it at the Bike Shop in Cerulean City, and you'll get a Bicycle you can use anywhere!