Becoming The Champ
One of your primary goals in X/Y is to become the Pokémon Champion . This is done by beating the Elite Four; four extremely talented trainers and then the current residing Champion. To be able to face them however, you must defeat the eight Gym Leaders in your region, the Kalos region.
They are spread out among various cities and you must travel to find, meet and challenge them. If you beat one, they present you with a Gym Badge. Having all eight Gym Badges grants you the right to face the Elite Four. This task isn't easy and will take many hours to achieve; but will progress naturally along with the rest of the storyline.
Gotta Catch Them All!
Besides wanting to be the very best, another key goal is catching as many Pokémon as possible, in a bid to complete your Pokédex. In Kalos alone, there are over 300 Pokémon and, throughout the six generations of Pokémon, there are slightly over 700 Pokémon in the National Pokédex, so it's by no means an easy feat...
However thanks to the wonders of the internet, it's now easier than ever to get the Pokémon you want. Just try to be smart with your trades (i.e. refrain from offering a lowly Pidgey for a legendary Arceus) and it shouldn't take too long to find your desired Pokémon.
Even if you don't intend to fill up your Pokédex, you'd still be wise to catch what you can, as the more Pokémon you have, the more interesting the game becomes and the more strategies you can employ. As fun as a Charizard solo sounds like, it will probably get boring fast!
Whatever You Want!
The best thing (or one of them) about the Pokémon games is the complete freedom that you have. Okay, your progress will be hampered if you don't go through the story in a timely manner, but besides that you can do whatever you want at your own pace.
This can include seeking out human opponents in exciting, challenging battles (so no more overpowering the CPU trainers), making your Pokémon super happy in Pokémon-Amie or perhaps breeding a fearsome party of Pokémon with amazing moves and enviable IV values.