Starter Pokemon

Kanto
In Pokemon Red, Blue, Green, FireRed and LeafGreen Versions, Professor Oak brings Trainers to his Lab to give them one of the three Kanto starters:

The player's Rival (Blue) will choose the starter with the tyoe advantage over yours. The remaining Pokemon will sit on Prof. Oak's desk for the rest of the game.

In Pokemon Yellow Version, Professor Oak instead gives out: This Pikachu attacked the player on Route 1, which Prof. Oak caught. The player's Rival recieves an Eevee.

Johto
In Pokemon Gold, Silver, Crystal, HeartGold and SoulSilver, Professor Elm has the player run an errand for him and gives him/her one of the three Johto starters. The player keeps the Pokemon after completing the task:

A theif (Silver) steals the Pokemon with the type advantage over yours. The theif then becomes your Rival.

Hoenn
In Pokemon Ruby, Sapphire and Emerald Versions, atfer Prof. Birch is rescued by the player from either a Poochyena (Ruby/Sapphire) or a Zigzagoon (Emerald). The professor gives the player the Pokemon they used to defeat the attacking Pokemon:

The player's Rival (either May or Brandon) will already have the starter Pokemon (given to them by their father, Prof. Birch). Their chosen starter will have the type advantage over the player's.

Sinnoh
In Diamond and Pearl, the player and the Rival are attacked by wild Starly while at Lake Verity. The player must choose one of the three Pokemon inside a nearby briefcase. One of:

The player's Rival will then choose the starter with the type advantage over the player's. This also marks the first game in the series in which all three of the starter's are taken. One by the player, one by the Rival and one by Dawn or Lucas.

Isshu
In Black and White, the Professor Araragi (the first female Pokemon Professor) will give the player one of the three Isshu starters: Tsutarja, Pokabu or Mijumaru:

Phuto
In Sun and Shadow, during the introduction segment of the game, Prof. Hawthorn will ask the player to choose a starter. When the main game begins, the player uses it to find the other two starters.

Simmons will later show the player how to battle. The person he battles has the starter with the type advantage over the player's and also becomes the player's Rival.

Orre
In Pokemon Colosseum, the player will recieve both an Espeon and an Umbreon. This is a first for the Pokemon games.

In Pokemon XD: Gale of Darkness, the player already has an Eevee at the beginning of the game. It is at Level 10:

At the beginning of the game, someone will offer the player one of the five evolutionary triggers (Water Stone, Thunderstone, Fire Stone, Moon Shard or Sun Shard), these items will evolve the Eevee into one of the five eeveelutions (at the time).

Pokemon Mystery Dungeon: Red and Blue Rescue Team
In Pokemon Mystery Dungeon: Red and Blue Rescue Team, the following Pokemon are avaliable as candidates for the Hero and Partner.

Pokemon Mystery Dungeon: Explorers of Time and Darkness
In Pokemon Mystery Dungeon: Explorers of Time and Darkness, the following Pokemon are avaliable as candidates for the Hero and Partner:

Pokemon Mystery Dungeon: Explorers of Sky
The following Pokemon are avaliable as candidates for the Hero and Partner:

Pokemon Mystery Dungeon: Keep Going! Blazing Adventure Squad!
In Pokemon Mystery Dungeon: Keep Going! Blazing Adventure Squad! The following Pokemon are avaliable as the Hero and Partner:

Pokemon Mystery Dungeon: Let's Go! Stormy Adventure Squad!
In Pokemon Mystery Dungeon: Let's Go! Stormy Adventure Squad! the following Pokemon are avaliable as the Hero or Partner:

Pokemon Mystery Dungeon: Go For It! Light Adventure Squad!
In Pokemon Mystery Dungeon: Go For It! Light Adventure Squad! the following Pokemon are avaliable as candidates for the Hero and Partner: