Collecting and assembling your own unique party of Pokémon is one of the many awesometastic things that you can do in any and all Pokémon games. At the very start of your Pokémon journey you will receive one of three starter Pokémon from the kind Professor Sycamore (well, from your friends from him).
At certain points in the game, you may receive free Pokémon like your initial starter, but don't expect a lot of these opportunities. Instead, you'll need to use your own hands and feet! What we mean is that you can obtain more Pokémon by catching them in the wild.
This is done by throwing Poke Balls at them, ideally when they're weak and about to faint. While you do this, beware of the power of your Pokémon's attacks, as dropping a wild Pokémon down to 0 HP will cause it to faint, preventing you from capturing it!
To make life easier, you can have an applicable Pokémon (one with sharp claws) use False Swipe , which can be learned early on - a move that can reduce an opposing Pokémon's HP down to 1 at most. It also helps loads if you inflict a status ailment on the opposing Pokémon too - Paralysis or Sleep are good, but not Burn or Poison in case they faint.
Although you can collect as many Pokémon as your PC can hold (which is a lot), you can only carry six around with you at any one time which form your party (these are the ones that you use in battle). Collecting and assembling a strong party is all a part of the fun!
Besides in-game freebies and wild Pokémon, you can also acquire Pokémon - especially Pokémon that cannot be found in your version of the game (or Pokémon X and Y in general) - by trading with other players. You can instigate a trade locally or online by using the PSS (Player Search System) on the bottom screen.
If you have no friends, you can still use the PSS to trade Pokémon via the GTS (Global Trade Station), where you can seek out trade requests or submit your own trade request, then wait until it's fulfilled by another player. Or if you're really feeling lucky, you can use Wonder Trade to instantly trade Pokémon with a random player.
Other Ways to Obtain
Finally, if you keep your eyes peeled in the right places, you can occasionally obtain rare and exclusive Pokémon from special events or similar avenues. For instance, from 12th October 2013 to 15th January 2014, you could receive a Torchic with Blazikenite by connecting to the internet and using the Mystery Gift option before loading your game.
Note : Refer to Tips and Secrets - Event Pokemon Distributions for further information.