Well I put this quick walkthrough together which tells you what your objectives are in each city so if you need help then you could look here. If I did a detailed one with pictures and stuff it would take up a lot of space so bare with me. At least its something.
Pallet Town
-Choose one out of 3 Pokemon from Professor Oak as your first Pokemon.
Route 1: Pallet Town to Viridian City
Viridian City
-Go to the Poke mart and pick up Prof. Oaks parcel from the clerk.
Route 1: Viridian City to Pallet Town
Pallet Town
-Give Prof. Oaks parcel in return for a Pokedex.
-Get a map from Gary’s sister
Route 1: Pallet Town to Viridian City
Viridian City
Route 2: Viridian City to Pewter City
Viridian Forest
Pewter City
-Defeat Brock at the Pewter City Gym for the Boulder Badge
Route 3:Pewter City to Mt. Moon
Mt. Moon
Route 4: Mt. Moon to Cerulean City
Cerulean City
-Defeat Misty in the Cerulean City gym for the Cascade Badge.
Route 24 and 25: Cerulean City to Bills House
Bills House
-Help Bill and get a ticket to go aboard the S.S. Anne in return
Route 24 and 25: Bills House to Cerulean City
Cerulean City
Route 5: Cerulean City to Saffron City
Underground Path:Cerulean City to Vermillion City
Route 6: to Vermillion City
Vermillion City
-Obtain HM01 from the captain of the S.S. Anne
-Defeat Lt. Surge of the Vermillion City Gym for the Thunder badge.
-Get the bike voucher from the Pokemon fan club
Route 11: Digletts Cave
Route 2: The other side of the trees
-Get HM05
Digletts Cave to Route 11 to Vermillion City to Route 6 to the Underground path and Route 5
Cerulean City
go to bike shop to get a bicycle with the bike voucher
Route 9: Cerulean City to Rock Tunnel
Rock Tunnel
Route 10: Rock Tunnel to Lavender Town
Lavender Town
Underground Path: Lavender Town to Celadon City
Celadon City
-Beat Erika in the Celadon City Gym for the Rainbow Badge.
-Obtain Silph Scope from Team Rocket Hideout beneath the Rocket Game Corner.
-Buy a beverage from the vending machine on the highest floor of the Celadon City Dept. store
Underground Path: Cerulean City to Lavender town
Lavender Town
-Rescue Mr. Fuji in the Pokemon Tower
-Get the Poke flute
Route 8: Lavender Town to Saffron City
-Give thirsty guard something to drink
Saffron City
-Rescue President of Silph Co. and receive the master ball
-Go to fighting Dojo and get Hitmonchan or Hitmonlee
-Defeat Sabrina in the Saffron City Gym for the Marsh Badge
Route 7: Saffron City to Celadon City
Celadon City
Route 16
-Wake up Snorlax with Poke flute
-Get HM02 from the house behind the bushes
Route 17 and 18: Cycling Road
Route 8: Saffron City to Lavender Town
Lavender Town
Route 12
-Wake up Snorlax with the Poke flute
Routes 13,14,15: The Docks
Fuschia City
-Defeat Koga of the Fuschia City Gym for the Soul Badge
-Pick up HM03 and gold teeth in the Safari Zone
-Give the gold teeth to the Game Warden in exchange for HM04
Route 19:Fuschia City to Seafoam Islands
Seafoam Islands
Route 20: Seafoam Islands to Cinnabar Island
Cinnabar Island
-find the secret key which opens the Cinnabar Island Gym in Pokemon House
-Defeat Blaine of the Cinnabar Island Gym for the Volcano Badge.
Route 21: Cinnabar Island to Pallet Town
Pallet Town and Route 1 to Viridian City.
Viridian City
-Defeat Giovanni, leader of Team Rocket, of the Viridian City Gym for the earth Badge
Route 22 and 23: Viridian City to Victory Road
Victory Road
Indigo Plateau
-Defeat the Elite Four
-Defeat Gary
-Become a true Pokemon Master