Contests are all about scoring points, which is hard to do as your opponents keep on doing attacks to startle your Pokémon. So a good way to get the upper hand is to do Combo Attacks.
A Combo attack is when you do two particular attacks in succession, the Judge will notice this and you will get a few extra points...which comes in handy. So here are all the combinations...if you wish to see the types of attacks please consult the Contest Attack List:

Starting Attack Secondary Attack(s)
Belly Drum Rest
Calm Mind Confusion
Dream Eater
Future Sight
Light Screen
Luster Purge
Meditate
Mist Ball
Psybeam
Psychic
Psycho Boost
Psywave
Reflect
Charge Shock Wave
Spark
Thunder
Thunderbolt
Thunderpunch
Thundershock
Thunder Wave
Volt Tackle
Zap Cannon
Charm Flatter
Growl
Rest
Tail Whip
Confusion Future Sight
Kinesis
Psychic
Teleport
Curse Destiny Bond
Grudge
Mean Look
Spite
Defense Curl Rollout
Tackle
Dive Surf
Double Team Agility
Quick Attack
Teleport
Dragonbreath Dragon Claw
Dragon Dance
Dragon Rage
Dragon Dance Dragonbreath
Dragon Claw
Dragon Rage
Dragon Rage Dragonbreath
Dragon Claw
Dragon Dance
Earthquake Eruption
Fissure
Endure Flail
Reversal
Fake Out Arm Thrust
Faint Attack
Knock Off
Seismic Toss
Vital Throw
Fire Punch Ice Punch
Thunderpunch
Focus Energy Arm Thrust
Brick Break
Cross Chop
Double-Edge
Dynamicpunch
Focus Punch
Headbutt
Karate Chop
Sky Uppercut
Take Down
Growth Absorb
Bullet Seed
Frenzy Plant
Giga Drain
Leech Seed
Magical Leaf
Mega Drain
Petal Dance
Razor Leaf
Solar Beam
Vine Whip
Hail Aurora Beam
Blizzard
Haze
Ice Ball
Ice Beam
Icicle Spear
Icy Wind
Powder Snow
Sheer Cold
Weather Ball
Harden Double-Edge
Protect
Rollout
Tackle
Take Down
Horn Attack Horn Drill
Fury Attack
Hypnosis Dream Eater
Ice Punch Fire Punch
Thunderpunch
Kinesis Confusion
Future Sight
Psychic
Teleport
Leer Bite
Faint Attack
Glare
Horn Attack
Scary Face
Scratch
Stomp
Tackle
Lock-On Superpower
Thunder
Tri-Attack
Zap Cannon
Mean Look Destiny Bond
Perish Song
Metal Sound Metal Claw
Mind Reader Dynamicpunch
Hi-Jump Kick
Sheer Cold
Submission
Superpower
Mud Sport Mud-Slap
Water Gun
Water Sport
Mud-Slap Mud Sport
Sand-Attack
Peck Drill Peck
Fury Attack
Pound Doubleslap
Faint Attack
Slam
Powder Snow Blizzard
Psychic Confusion
Future Sight
Kinesis
Teleport
Rage Leer
Scary Face
Thrash
Rain Dance Bubble
Bubblebeam
Clamp
Crabhammer
Dive
HydroCannon
Hydro Pump
Muddy Water
Octazooka
Surf
Thunder
Water Gun
Water Pulse
Water Sport
Water Spout
Waterfall
Weather Ball
Whirlpool
Rest Sleeptalk
Snore
Rock Throw Rock Slide
Rock Tomb
Sand-Attack Mud-Slap
Sandstorm Mud Shot
Mud-Slap
Mud Sport
Sand Tomb
Sand-Attack
Weather Ball
Scary Face Bite
Crunch
Leer
Scratch Fury Swipes
Slash
Sing Perish Song
Refresh
Sludge Sludge Bomb
Sludge Bomb Sludge
Smog Smokescreen
Stockpile Spit-Up
Swallow
Sunny Day Blast Burn
Blaze Kick
Ember
Eruption
Fire Blast
Fire Punch
Fire Spin
Flame Wheel
Flamethrower
Heat Wave
Moonlight
Morning Sun
Overheat
Sacred Fire
Solarbeam
Synthesis
Weather Ball
Will-O-Wisp
Surf Dive
Sweet Scent Poisonpowder
Sleep Powder
Stun Spore
Swords Dance Crabhammer
Crush Claw
Cut
False Swipe
Fury Cutter
Slash
Taunt Counter
Detect
Mirror Coat
Thunderpunch Fire Punch
Ice Punch
Vice Grip Bind
Guillotine
Water Sport Mud Sport
Refresh
Water Gun
Yawn Rest
Slack Off

 
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