The Power Plant can be reached from Route 10 (via Cerulean City) after you learn the Secret Technique Sea Skim in Fuchsia City.
How to Get to the Power Plant
After you learn Sea Skim in Fuchsia City, go to Cerulean City. Go east from there to Route 10. Surf in the water and follow the path of the water until you reach the end. This is where you will find the Power Plant.
At the Power Plant
Outside, east of the Power Plant building, you can battle Poké Maniac Mark, who has a level 42 Rhyhorn and a level 42 Lickitung. You get 2016 Pokédollars and 3 Poké Balls for winning.
West of the Poké Maniac, there is a Thunder Stone next to the Power Plant building.
Explore the Power Plant
Go into the Power Plant. In here, you can catch Voltorb, Magnemite, and others.
Go north from the entrance and look to the right to find a Max Revive.
To the north, you can battle Coach Trainer Mable, who has a level 44 Omastar, a level 45 Aerodactyl, and a level 44 Kabutops. You get 4500 Pokédollars for winning. After the battle, you get 5 Rare Candies.
Go east from here, then southeast, then east. There are two paths going east. Use the upper of these two paths. Go east and check on the last red barrel in the southern row of barrels to find a hidden Voltorb Candy.
Keep going east and check on the last red barrel in the northern row of barrels to find a hidden Voltorb Candy.
Go east, then north, to find a Max Potion.
Go west along the northern path, and check the first red barrel to find a hidden Voltorb Candy.
Go west. The item that looks like a Poké Ball is actually an Electrode. Check on it if you want to try to catch it (you will have to defeat it within 5 minutes first).
Go west and check on the last red barrel to find a hidden Voltorb Candy.
Then go east, then south, then west to leave this area, then go south when you can. Go east to find an X Sp. Def.
Go southeast from there, then go southwest. In this room, go east to find a Rare Candy.
Then go west, then north. Then go east, then south when you can, to find TM38 Thunder.
You can go to the east side of this room to find another Electrode that you can try to defeat and catch.
Then go north out of this room. Go east, then north up a hallway. Then go into the room to the north. In the northeast of this room, there is a Max Potion, and in the northwest of this room, there is a Thunder Stone.
Go south out of this room, then go west and follow the path. Along the way, you will find another Electrode that you can attempt to defeat and catch.
Go south from that Electrode, then check the blue chair on the right to find a Pretty Wing.
Go south from there into a room where there is a Paralyze Heal. There is also an Electrode in the corner that you can try to defeat and catch.
Go north out of this room, then west. If you are ready to try to defeat and catch a Legendary Pokémon, go north. (If you are not ready, you can use the teleporter in the southwest corner of this room. Be sure to have plenty of Ultra Balls.)
Zapdos
Save your game, go north to battle Zapdos. You have five minutes to defeat it. If you don't defeat it in time, it will run away. To make it come back, you will have to defeat the Elite Four. (Or you can just close the game without saving, and try again.)
Once you defeat it, use Ultra Balls (I don't recommend using your Master Ball) and use Razz Berries to try to catch it. If it runs away, you have to defeat the Elite Four to make it come back. (Or you can just close the game without saving, and try again.)
Afterward, you can go southwest and step on the teleporter to go outside.
Moving On
You can go south from Fuchsia City and go through Route 19 and Route 20 to reach the Seafoam Islands, where another rare pokémon can be found.
You can surf in the water in Pallet Town and go south from there through Route 21 to reach Cinnabar Island, where the next Pokémon Gym awaits.