Is there a order to play Pokémon?
No, Pokémon is a game that you can play ANY Pokémon game without playing his predecessor. So, pick one of the games of your choice (game-recommendation is off the chart. go to the chat, there will be more help about it).
The story doesn't have continuation?
Yes and No... Yes, there is a continuation. No, the story is not continuated IN-GAME, as @Kendra pointed out in her comment
What's the advantage on playing the predecessors?
You can "migrate" the Pokémon that you captured to the new game.
Do I need to play the old games to get all Pokémon?
No. You can find them at the Global trade.
The games appears to come in pairs, what's the meaning of this?
The games comes in pairs to create a more strong structure. Usually the "Main Pokémon" (the one in the cover) is different depending on the game, and there are exclusive Pokémon too. Even exclusives zones.
The idea is that you play one, and trade with someone that plays the other. So you can Fill up your Pokédex.
Does the console matter?
Yes, you can only play some of the Pokémon games in determinate consoles, also. Global/Local events are available only in some specific games (and as you can't play Pokémon Black 2 in your gameboy advanced or yellow in your 3dsxl ... you get my point)
But, besides that, No. The console doesn't matter.