For the X&Y series, I'd include the following characters as rivals:
Calem - Ash's main rival in the series. Initially he is too hard on both himself and his pokemon, but Ash teaches him to calm down. Later it is revealed that Calem has seen all of Ash's league battles on TV and pushes his pokemon hard in an effort to be as strong as he sees Ash to be.
Chespin > Quilladin > Chesnaut - Calem's starter pokemon is also its powerhouse, a reputation earned time and again as its slams down opponent after opponent with Wood Hammer, Brick Break, Spiky Shield and Dragon Claw
Litleo > Pyroar - Timid at first, Litleo soon learns that its fire is its true power, something which proves even more true once it evolves. Knows Flamethrower, Hyper Beam, Noble Roar and Take Down
Poliwag > Poliwhirl > Politoed - A surprising capture that soon proves its worth. Its Drizzle ability gives it the edge in any battle. Knows Bubble Beam, Ice Beam, Power-Up Punch and Focus Blast
Snorlax - Calem catches this sleeping pokemon that was blocking a bridge as a sign of his admiration of Ash. He teaches his Rest, Sleep Talk, Heavy Slam and Wild Charge
Eevee > Jolteon - (here because I think it needs more screen time, but also as a nod to the games) This is Calem's counter to Pikachu. It knows Thunder, Pin Missile, Quick Attack and Shadow Ball
Lucario - A gift from Korrina, this is the first pokemon in the anime, aside from Blaziken, to be seen Mega-Evolved. Knows Close Combat, Extreme Speed, Swords Dance and Aura Sphere.
Arbok - (I just want to see one that can put up a good battle. Enough time has passed since Jessie's left, and we've seen a decent Koffing since then!) Knows Crunch, Mud Bomb, Toxic and Venoshock
Scyther > Scizor - Calem's second pokemon capable of Mega-Evolving is also his answer to Ash's Mega-Heracross. Knows Bullet Punch, X-Scissor, Aerial Ace and Swords Dance
Noibat > Noivern - This is Calem's answer to Ash's Charizard. Then 'zard Mega-Evolves... Knows Dragon Pulse, Boom Burst, Hurricane and Dragon Claw
Shauna - Mainly Serena's rival. Often underestimated because of her team of cute pokemon, she is actually more powerful than people are prepared to give her credit for. A lot of her pokemon rely on healing moves.
Goomy > Sliggoo > Goodra - Shauna's first pokemon is a lot like her - easily understimated. However, once it evolves and gets the moves Dragon Pulse, Body Slam, Muddy Water and Aqua Tail under its belt it proves to be a very formidable opponent, especially when its Sap Sipper ability kicks on against unsuspecting grass types.
Skitty > Delcatty - (a nod to the games here) This is what May's Skitty failed to be! It knows the attacks Feint Attack, Double-Edge, Heal Bell and Charge Beam
Flabebe > Floette - The smallest of Shauna's pokemon is also the most underestimated in battle. Knows Moon Blast, Petal Blizzard, Vine Whip and Aromatherapy
Furfrou - Shauna's Furfrou is not the first pokemon to want to be in a fight, but it sure as hell won't back down from one once its in it. Ash swears the thing has a differnt haristyle everytime he sees it. Attacks are Wild Charge, Headbutt, Refresh and Dark Pulse
Eevee > Flareon - Rather than have a Sylveon, which went to Serena, Shauna gets a Flareon instead. This fire type is one hell of a fire breather - so long as her coat doesn't get too dirty! Knows Flare Blitz, Fire Blast, Shadow Ball and Wish
Mareep > Flaaffy > Ampharos - Ampharos is the only pokemon on Shauna's team capable of Mega-Evolving and its doing so usually turns the tide of battl in her favour. It knows Thunderbolt, Thunder Punch, Dragon Pulse and Light Screen
Cleffa > Clefairy > Clefable - Recieved as an egg Cleffa wastes no time in proving its power, which only increases as it evolves. Knows Moon Blast, Focus Blast, Meteor Mash and Moonlight.
Lapras - A gift from a man you couldn't take care of it anymore, Lapras is Shauna's water transport and in battle, is what Ash's Lapras never got the chance to be! Knows Hydro Pump, Ice Beam, Body Slam and Sing
Trevor - Another rival of Ash's. Trevor feels that by seeing every pokemon listed in the pokedex he'll have the knowledge needed to become the best trainer in the world.
Pikachu > Raichu - Trevor's first pokemon initially gets along well enough with Ash's Pikachu. Then evolves. Ash is no impressed that Raichu knows Focus Blast, a move that Pikachu could never hope to learn in its current form, that Pikachu is driven to consider the possibility of evolving once again. Raich knows Thunder, Focus Blast, Grass Knot and Light Screen.
Aerodactyl - Aerodactyl is revived from a fossil at the same time as Ash's Tyrunt and Serena's Amaura. It is capable of Mega-Evolution, and is a dangerous opponent when it does. It knows Crunch, Sky Drop, Ancient Power and Supersonic.
Pumpkaboo > Gourgeist - This spooky pokemon uses its Trick-or-Treat attack to make its own moves hit the opponent harder. Also knows Phantom Force, Shadow Ball and Razor Leaf
Croagunk > Toxicroak - Trevor's specialist in hand-to-hand combat, its poison type moves are not to be sniffed at. Knows Low Sweep, Power-Up Punch, Sludge Bomb and Feint Attack.
Clauncher > Clawitzer - This is Trevor's counter to Ash's Frogadier. It has the Mega Launcher ability and its attacks, Water Pulse, Dragon Pulse, Dark Pulse and Aura Sphere make full use of it.
Diggersby - Clemont's Bunnelby doesn't evolve. Instead Diggersby's role in the series comes from Trevor. It knows Hammer Arm, Bulldoze, Dig and Swords Dance
Lairon > Aggron - This is the second pokemon Trevor has that is capable of Mega-Evolving. Its attacks are Heavy Slam, Iron Defence, Dragon Claw and Stone Edge
Heatmor - Trevor's last pokemon is Heatmor, a pokemon capable of burning up any battlefield and anything on it. It knows Inferno, Hone Claws, Slash and Focus Blast
Tierno - A rival to Ash, but one who is more like Stephan. He is convinced that the best way to battle is to well-practiced in the art of dance, and his pokemon have moves that reflect this.
Corphish > Crawdaunt - Tierno's first pokemon is a tough one to battle, especially one it has a Swords Dance or two under its belt. Also knows Night Slash, Vice Grip and Scald
Roselia > Roserade - This pokemon is such a good dancer that it can be very hard to land a hit on her in battle. Knows Petal Dance, Weather Ball, Poison Jab and Solar Beam
Hawlucha - (I couldn't bring myself to give him Talonflame as Ash has one, so this is a good second) A tricky aerial battler that further complicates things by wrestling and dancing. Knows Aerial Ace, Acrobatics, Flying Press and Swords Dance
Spinda - A tricky little pokemon that relies on its Teeter Dance to confuse its opponents before hitting them with Psybeam, Feint Attack and Shadow Ball.
Mr Mime - This one not only dances, but creates defensive walls as well. Knows Teeter Dance, Barrier, Psychic and Magical Leaf.
Jynx - Tierno's last pokemon is a bit of a surprise, but it's singins serves as the rhythm that the others learn to dance to. Knows Sing, Psyshock, Ice Punch and Blizzard.
Blaziken Trainer - (whatever his name is) Not initially a rival, this changes at the league where Ash ends up facing his MegaBlaziken with his MegaCharizard for a Zard-Ken rematch.
Blaziken - The main pokemon used by this man has the speed boost ability to really give its oppoents trouble. Knows Flare Blitx, Hi Jump Kick, Bulk Up and Shadow Claw
Druddigon - (because this one needs more screen time) This one hits hard and fast. Knows Dragon Claw, Dragon Tail, Super Power and Flamethrower
Tyranitar - (because when has Ash ever battled one of these?) Capable of Mega-Evolving, but usually doesn't in favour of Blaziken. Still powerfull though, knowing Stone Edge, Thunder Fang, Focus Blast and Shadow Claw
Trevenant - This ghostly battler uses Ghost and Grass moves to get the edge in any battle. Knows Horn Leech, Phantom Force, Will-O-Wisp and Leech Seed
Aegislash - Switching between its two forms repeatedly throughout the battle causes frustration in many opponents. Knows King's Shield, Sacred Sword, Night Slash and Iron Head
Starmie - (mainly here so we can see a good one, rather than the neglected one that Misty had) Uses a combination of moves to confuse, paralyse and stay in battle as long as possible. Knows Confuse Ray, Thunder Wave, Recover and Hydro Pump