You will encounter team rocket after you beat giovanni, but you will never encounter giovanni in the game again.
To the neighbourhood
when you beat Giovanni in the game corner
first u see him in Celadon City, in the basement of the Rocket Gambling area. When you beat the Rocket Grunt at the back, he leaves and the switch is in the poster he was looking at. After you've beaten Giovanni there, he moves on to the Silph Co. in Saffron City. He is holding the Silph Co. President hostage, and you have to get through the company maze and then beat Giovanni there. Once you do that, he goes to Viridian City and goes back to the gym.
he is in the Pokemon tower you have to beat Giovanni in the arcade in ceadon city then you get the sliph scope to look a ghosts
You will encounter team rocket after you beat giovanni, but you will never encounter giovanni in the game again.
To the neighbourhood
The serious last time you beat Giovanni is at the viridian city Pokemon gym.
After you beat Giovanni in Pokémon Red, Blue and Yellow he goes to the gym in Viridian City. He is the final gym leader.
As the gym leader, Giovanni is supposed to be the eighth of them. So you have to beat the other seven before you can get to him.
beat blaine at cinnibar and then Giovanni at viridian city.
nowhere
when you beat Giovanni in the game corner
you must beat the other 7 gyms
You have 2 beat the elite 4 in Saffron City............ when you have beaten them you face your rival on the floor where you get 2 beat Giovanni... horrayyy you beat your rival...then beat Giovanni .... Afterwards.. you c the owner of sliph co. he gives you the master ball...
Earthquake (TM26) is your reward after you beat Giovanni
You have to beat Giovanni in the Celadon City Gym.