Strategy:

This is the standard Special sweeping set. Calm Mind up and hit opponents with Psychic. Thunderbolt gives you extra coverage against opponents that may be resistant to Psychic.
Gardevoir also has access to a couple of status moves that can be handy in crippling opponent. Will-o-wisp is useful for covering your weak Defense by inflicting a nasty Burn on physical attackers. Hypnosis is less accurate, but it's an effective way of disabling a potential threat if you can get it to hit.

Of Gardevoir's two abilities (Trace and Synchronize), Trace is the better choice. While Synchronize may be handy for automatically inflicting a status back onto the opponent, Gardevoir does not like to be statused in the first place since it decreases its ability to survive. Meanwhile, you can copy all sorts of useful abilities if you have Trace. Volt Absorb, Water Absorb, Levitate, Speed Boost, even Wonder Guard can all be obtained with this one cool ability.

Timid nature along with 200 Speed EVs gives you 270 Speed total. This is important for outrunning Adamant Heracross, who is a major threat to most Psychic types (Jolly Heracross and those holding Choice Scarf will still beat you though). The HP EVs help you reach 301 HP, which gives you a chance against Blissey by letting you survive up to 4 of her Seismic Tosses. Put the rest of the EVs into Special Attack so that it will have plenty of power even without using Calm Mind.

Choice Specs

Other options: Reflect, Focus Blast, Thunder Wave, Toxic, Energy Ball/Grass Knot, Destiny Bond, Taunt, Snatch. Gardevoir has a very large and diverse movepool, so it has plenty of options to work with and you can mix and match these moves with the ones already mentioned.

Reflect is an option over Will-o-wisp. It does the same job of cutting physical damage in half and though it's only temporary, you can switch and pass the benefits along to other members of your team.

Getting a particular Hidden Power is hard in-game, so Focus Blast is an option over Hidden Power FIghting for a stronger, less accurate Fighting move.

Thunder Wave is yet another one of the many status moves that Gardevoir can learn. It's a good move that can cripple sweepers who rely on their speed. Toxic is there because it's a great way of defeating Wobbuffet, who has the Shadow Tag ability that you may want to Trace. Just poison Wobbuffet and use Calm Mind while Wobbuffet can't escape or harm you at all, eventually dieing to Toxic damage.

Grass moves like Energy Ball and Grass Knot can deal massive damage to Tyranitar and Rhyperior, as well as OHKO Swampert. Grass Knot is better for the heavier Pokemon mentioned, while Energy Ball does more consistent damage.

Many people don't expect Destiny Bond coming from a Gardevoir and it could be nice for a surprise, especially against tough-to-beat opponents. Taunt messes up walls like Blissey and Snatch lets you steal all sorts of useful things like stat boosts, Substitutes, and healing moves.

 

How to counter Gardevoir:
Blissey is the most reliable counter against Gardevoir. She takes almost nothing from its special attacks and has Natural Cure/Aromatherapy to heal any status effects. Gardevoir could potentially defeat Blissey if it builds up enough Calm Minds, but it will have a hard time doing that when getting hit with constant Seismic Tosses. Snorlax and Regice are two other special walls that can do well against it. Snorlax can hit it with powerful physical attacks like Return and Crunch, but hates getting burned by Will-o-wisp. Regice can copy its special boosts with Psych Up and discourage it from using Calm Mind.

Houndoom is one of Gardevoir's biggest threats. It switches in easily since it is immune to both Psychic and Will-o-wisp. Be careful not to let Gardevoir get two or more Calm Minds in, or else it can seriously hurt Houndoom with Thunderbolt and it can also OHKO with Hidden Power Fighting.

Heatran resists Psychic, is also immune to Will-o-wisp and can deal major damage with Fire attacks coming from its excellent Special Attack. Calm Minds can lessen the pain of these attacks, but without a recovery move, it will eventually fall. Be sure not to let Gardevoir Trace its Flash Fire or you'll be out of luck.

Celebi resists its two main moves and like Blissey, can easily cure itself of any status effects. It can also annoy it with Leech Seed and use a Calm Mind of its own to set up against Gardevoir.

Basically hitting Gardevoir with physical attacks, especially supereffective ones, is a good idea. It's even better if they are immune or resistant to Psychic. Weavile easily outspeeds it while Tyranitar shuts off its valuble Leftovers recovery with Sandstream. However, all non-Fire physical attackers risk getting Burned by Will-o-wisp. Since it can learn so many different moves, it may also have a supereffective attack against the Pokemon you're using to counter it.

Finally, Choice Scarf Heracross gets a special mention. With Guts, it can absorb an incoming Will-o-wisp and using Choice Scarf, it easily outspeeds it to KO with Megahorn. Of course, it's weak to Psychic which is why good prediction skills are necessary.

 

Links to Pokedex Entries

Bulbapedia
LegendaryPokemon
Psypoke
Serebii
Smogon
Veekun