Pokémon X and Y Walkthrough, Pokémon Move Sets: Garchomp
(Please note that the recommendations below are largely made for in-game play. A Garchomp in a competitive environment will likely make use of different moves to accommodate for smarter, more adaptive opponents. If you have a different Garchomp build in mind, toss it into the comments.)
Though the fourth generation of pokémon had a few questionable inclusions, Garchomp wasn't one of them. Easily one of the most popular dragon-types of all time, as well as a staple of competitive play, Garchomp can shred the majority of pokémon with its combination of powerful moves and extremely-strong stats. You don't want to face down a Garchomp unprepared... especially if it's carrying Garchompite.
Type: Dragon / Ground
A three-stage pokémon, Garchomp begins as the diminutive Gible. Level Gible to 24 and you'll receive a Gabite; double this level to 48 and Gabite will evolve into a full-fledged Garchomp.
Fire, Poison, Rock. Immune to Electric.
Dragon, Fairy, Ice (4x).
Garchomp is an absolute beast almost across the board. Its top stat is Attack at a towering 130, allowing Garchomp to rip most enemies apart. HP comes next at 108, then Speed at 102. Defense and Special Defense are both above average, though neither are overwhelmingly good. Special Attack comes in last, and, really, that's not a problem for Garchomp anyway.
Mega Garchomp, by contrast, receives an enormous boost to its Attack, up to a base 170. That's just disgusting. Everything else rises at least a little bit save Speed, which drops by 10 points (but is still pretty awesome).
Sand Veil: When deployed during a sandstorm, this pokémon receives a boost to its evasiveness. Not bad if you have a pokémon with Sand Stream on your team, but Sand Veil really isn't that great on Garchomp alone. (That said, it has more use now that Garchomp can turn into Mega Garchomp.)
Rough Skin: Any enemy pokémon attacks that make physical contact will deal slight recoil damage to the attacker. Garchomp usually gets targeted by special moves, but physical attacks aren't that uncommon. I prefer Rough Skin over Sand Veil. This is a hidden ability.
Sand Force: Ground-, rock- and steel-type moves receive a significant boost to attack power during a sandstorm. So long as you think your Garchomp can survive a round to deploy Sandstorm after changing into Mega Garchomp, this move can be ridiculously potent. Sand Force is only available to Mega Garchomp.
There are few pokémon as perfectly suited to sweeping as Garchomp, and it's kind of wasted in any other role. If you only intend on using a normal Garchomp, you'll want to fill up all four slots with strong offensive moves that cover a wide range of pokémon. Outrage is Garchomp's number one STAB weapon against most opponents, save fairy- and steel-types, while Earthquake covers most other vulnerabilties. Past that you can gift Garchomp with moves that will cover holes in your team's coverage, such as Brick Break, Aerial Ace, Poison Jab, Crunch, Fire Fang, Stone Edge / Rock Slide, and Return. You can toss Swords Dance in if you want to ensure knockouts, though in most cases this will be simple overkill. (Competitive play is another matter, of course, as those pokémon have much higher defenses than the ones you face in-game.)
Intending to turn your Garchomp into Mega Garchomp? You'll want to make use of Sand Force. Remove one of your non-STAB moves and toss in Sandstorm. Use it upon transforming into Mega Garchomp and Earthquake / Stone Edge / Rock Slide will become ridiculously powerful, allowing you to cover the vulnerabilities of a massive number of pokemon.
EV / Super Training
A normal Garchomp build should focus on building up Attack and Speed. Construct your Garchomp in such a way that your opponents won't stand a chance of going first, if at all. If you want to go the Mega Garchomp / Sand Force route instead, emphasize Attack and Special Defense. The latter will make it more likely that Garchomp can ward off ice-type moves, its greatest weakness by far.
Catching a Garchomp
Garchomp itself doesn't appear in the wild, but you can find a wild Gible. They appear with fair regularity on Route 13 as moving clumps of sand. Grab one and start leveling.