Pewter City Gym
Petwer City Gym's Brock uses rock-type Pokémon Onix and Geodude. Since rock-Pokémon are immune to Pikachu's electric attacks, you'll need to capture more Pokémon on your way to Pewter City. Capture a Mankey on Route 22 as rock-type Pokémon will easily succumb to Mankey's fighting-type attacks. Evolve a Metapod to a Butterfree if you want extra support.
Cerulean City Gym
Misty uses two water-type Pokémon, Staryu and Starmie. Given water's weakness to electric attacks, you should have no problem bringing Misty down solely with Pikachu. Mankey and Butterfree should serve as decent support. After you defeat Misty, head north to cross the Nugget Bridge. Battling and defeating its trainers will earn you a gold nugget and a Charmander.
Vermilion City Gym
Lieutenant Surge uses an electric-type Raichu, which is weak to fighting-type and rock/ground-type attacks. Use your Mankey against this Pokémon, but also consider bolstering your party with a Geodude from Mt. Moon or a Sandshrew from Route 4. Afterward, talk to the police officer in the middle of the city and she'll give you a Squirtle as reward for defeating Surge.
Celadon City Gym
Celadon City Gym's Erika utilizes a team of grass-type Pokémon, consisting of Weepinbell, Gloom and Tangela. Train the Charmander you received outside of Vermilion city until it evolves to Charmeleon, and you shouldn't have much problem dispensing of Erika. Consider adding a Pidgeotto or Fearow (evolved from Pidgey or Spearow, found on most routes) to bolster your party.
Saffron City Gym
Sabrina's team of psychic-type Abra, Kadabra and Alakazam may prove a challenge to your party. Psychic-type Pokémon have no real weakness, and they are strong against fighting- and poison-type Pokémon. Psychic Pokémon have low defense, though, so you should be able to take them down with a party consisting of Pokémon with high special attack stats. Consider Blastoise (evolved from Squirtle, then Wartortle), Charizard (evolved from Charmander, then Charmeleon) and Pikachu.
Fuchsia City Gym
By this point, Koga's team of three Venonats and a Venomoth shouldn't even give you pause. You can easily obliterate all four Pokémon with Charizard or any other fire-type Pokémon. You can also use Pikachu or any rock/ground-type Pokémon to take out Venomoth.
Cinnabar Island Gym
Blaine uses fire-type Arcanine, Rapidash and Nintetales. A high-level Blastoise should easily take care of all three, but you could round out your party with a rock or ground Pokémon such as Graveler or Sandslash.
Viridian City Gym
Giovanni utilizes an eclectic team consisting of ground-type Dugtrio, normal-type Persian, poison-type Nidoqueen and Nidoking, and rock-type Rhydon. Use Primeape (evolved from Mankey) or another fighting-type to take out Persian. Blastoise can handle Dugtrio and Rhydon. Use Charizard to take out Nidoqueen and Nidoking. Round out your party with a psychic-type such as Kadabra for support.