Jimmy > Gold > Male Trainer
Gold is the main character of the game and the male trainer of the Gold/Silver/Crystal era. Gold has the same goal as many other trainers which is to become a Pokémon Master and collect all 16 badges across Johto and Kanto.
Crystal > Female Trainer
In Pokemon Crystal the feature where you can choose between being a male or female was introduced. Crystal was based off the character of the manga but to play with the story of the game she has the same goal as the male trainer. (Gold)
Silver > Rival
Silver is different from rivals in the other games. They're usually friendly or teasing when Silver is unfriendly and mean. He stole his starter Pokémon from Professor Elm. He is also rude to his Pokémon and thinks all weak Pokemon should be left behind. One explanation for all this is that Silver is the son of Giovanni; The Leader of Team Rocket. It was revealed in the manga and in FireRed/LeafGreen.
Professor Elm is the Pokémon Researcher of the Johto Region. He gives you a starter Pokémon in return for you to go to Mr Pokémon's house to find out what his 'amazing discovery' is at the start of the game.
Professor Oak is the Pokémon Researcher of the Kanto Region. He is the one who gives you the Pokédex at the start of your journey when you visit Mr. Pokémon to retrieve his 'amazing discovery' for Professor Elm.
Your Mom will save up your money for you while you are traveling if you ask her, and will occasionally buy you useful items.
The man who discovered the first Pokémon Egg. He lives up north of Cherrygrove City.
Kurt will make special types of Poké ball for you once a day if you give him an Apricorn.
Eusine (Crystal Version Only)
A mysterious trainer that is on the hunt for the Legendary Pokémon Suicune. You will have to battle him multiple times throughout the game.
The champion of the Pokémon League. He helps you out in certain parts of the game.
Red is by far the strongest pokemon trainer in the game. He resides atop the formidable Mt. Silver in Kanto, silently awaiting his next challenger.