What is the fastest way to get to Fuchsia city?

From what I've seen from speed runners and my own experience, you only need to actually beat 3 gyms to reach Fuchsia City so a few recommendations are:

• Catch a full team as soon as possible (they will always get experience from battles and catches, a recommended team would be your starter, the first 2 bug Pokemon Weedle and Caterpie, Pidgey, the Magikarp sold to you before Mt. Moon, and a Geodude to beat Lt. Surge Gym easily).
• Play with 2-controllers to allow 2 vs 1 battles.
• Avoid some in-game trainers, not too many so you won't be under leveled.
• Also to move faster, when you reach Vermilion City there is an NPC just below Pokemon Center (a karate guy in LGP/beauty in LGE) that will ask you which Pokemon you like the most Meowth or Growlithe, it doesn't matter which one you choose, he will give you the evolved form depending on your game version, Persian for LG Pikachu, and Arcanine for LG Eevee, you can switch it for any of your Pokemon in your Party, take it out of the pokeball and you will ride it, this helps to move a lot faster in the game.

Edit: Forgot to say that the Karate guy will ask you to catch 5 Pokemon of your version exclusive (Meowth in LG Eevee and Growlithe in LG Pikachu), but they are really easy to find, just go north of the city and in those grasses it should spawn.

Now related to a quick strategy to get there faster is after reaching Lavender town, complete that story part ASAP:

• Go inside the tower until the little Cubone leaves.
• Exit town to the left to find Team Rocket and follow them to Celadon City.
• Go to the Casino and finish the Team Rocket Hideout.
• Go back to Lavender Town now with the Silph Scope and get the Pokeflute from Mr. Fuji
• Now go back to Celadon City and use the Pokeflute with the Snorlax blocking the path to route 16.
• Once you capture it or run away you're pretty much done, since you can pretty much avoid every trainer in route 16, 17 and 18 to reach Fuchsia City

I totally encourage you to take this routes, since the trainers are lower leveled compared to others routes to reach Fuchsia.