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GEN 1 - GameBoy Classic - Red, Blue, Yellow
GEN 2 - GameBoy Color - Gold, Silver, Crystal
GEN 3 - GameBoy Advance - Ruby, Sapphire, Emerald, FireRed, LeafGreen
GEN 4 - DS - Pearl, Diamond, Platinum; HeartGold, SoulSilver
GEN 5 - DS - Black, White, Black 2, White 2
GEN 6 - DS - X, Y {Not yet released}
Gen 1/2 games are not forward compatible - the continuity begins with Generation 3. That's not to say the early games aren't fun and viciously addictive; you just can't xfer your beloved Pokemon from those generations forward.
Each generation consists of a pair of games, released simultaneously, and a third game, released after the initial pair, including additional features (in bold), as well as an additional pair, a reboot of former generations (in italics). Each Pokemon only appears in two of the three games of each generation.
Each of the games of each pair or trio are essentially the same game, with the same gameplay; the only difference is the selection of Pokemon. Therefore, most generations have five versions of two games.
The third game in each trio largely makes the initial pair obsolete. Therefore, it's best to get Yellow, Crystal, Emerald, Platinum or B/W 2, and not bother with the initial pair.
Each generation is independently playable and the multiplayer game is centered on the latest generation, so whether you want to play them all or only the latest is a matter of preference.
