Pokémon FireRed and LeafGreen/Cerulean City

There is much to do in Cerulean City. You will have to battle your Rival, beat a rain of trainers, stop Team Rocket from their usual deeds and of course earn another badge. The way out is blocked by a guard, who will move away after you visit Bill at his house on Route 25.



Rival battle #3
When you go north, attempting to go on the bridge to Bill's house, your Rival will stop you for a fight. After the battle has ended, he will give you a Fame Checker.

Your rival can be quite tough to beat, even tougher than the Gym Leader of this city. His Pidgeotto is the toughest, it uses annoying Sand Attack which decreases Accuracy and forces to switch. His Rattata uses mainly Hyper Fang (very powerful attack in this stage) and Quick Attack. His Abra is very easy to beat since it can only Teleport. Make sure you have well enough trained Pokémon before you face him.

Gym battle
You might want to wait with the gym battle after you've visited Bill, since Misty can be a tough opponent at this point of the game. After you beat Misty you will be able to use the HM01, outside of battle.

If you have a Grass- or an Electric-type Pokémon in your team, this gym shouldn't be too difficult. An Ivysaur or well-trained Pikachu should be enough to beat Misty. If you don't have neither of these, save this gym for later when you have more better-trained Pokémon.

Starmie is the toughest one to beat, it has very powerful stats at this point and its Water Pulse (in addition to doing decent damage, it has a chance of confusing your Pokemon) packs a punch. Use Leech Seed and Razor Leaf if you have Ivysaur or Thundershock/Thunderbolt if you have Pikachu. Starmie can also restore a portion of its HP, so be prepared for a lengthy battle and ensure your Grass/Electric type Pokemon is at least level 20.

Starmie is somewhat easier to defeat if you can knock it down while its still using Water Pulse. After that, it turns to Swift, a highly damaging attack.

Trade
At the house west to the PokéCenter there is an old man who will trade Pokémon with you. You will get a Jynx at the same level as the Poliwhirl you have given. You won't be able to do this trade until much later in the game, after you get a Good Rod.

Bike Shop
You may buy a bike from the shop at the southwest corner of the city. However, it costs 1,000,000, which is more money than you are capable of holding. In Vermilion City, you will receive a Bike Voucher that you can exchange in this store for a free bike.

Leaving Town
When you finally leaving Cerulean from the house where the guard used to be, a Team Rocket Grunt will stop and fight you. When you defeat him he will give you a TM28.

Items

 * Rare Candy (hidden behind the house on the north west corner of the city between two flowers)
 * TM03 (given by Misty after you beat her)
 * Powder Jar (given by the old man in the 2nd upper house, but only if you already have at least one berry)
 * Fame Checker (given by your Rival after you beat him)
 * TM 28 (Given by the TR Grunt after you beat him)