If I may be frank, Pokémon X and Y are among the best (if not the best) games to do a Monotype (or Single Type) Run.  Pick almost any type and you’re guaranteed a fantastic time, even Ice and Dragon are doable to a degree.  With over 400 Pokemon available in these games lot of types neutralize their weaknesses and can be caught before the first gym.  So what would your team look like? Let’s take a look but first…

Rules

1.   You can only catch and train Pokémon of a certain type.  Once you capture your first Pokemon of that type, disregard any previous Pokemon on your team.
2.   If a Pokémon evolves to said type you may train it but evolve it ASAP.
3.   No trading allowed
4.   If a Pokémon changes to said type via Mega Evolution you are allowed to train it provided you Mega evolve it every time.
5.   Only Pokémon caught before the Elite Four are counted.

The Best Types

To give a cop-out answer, almost all the types in these games are great for a Monotype Run.  I’m serious, pick a type and you’re bound to have a good time.  But what if you want the very best? In my opinion, the best of the best are Flying, Water, and Fighting.  What these three types have in common are high diversity of strong Pokemon.  You can have multiple variations of each type and easily cover their weaknesses.  The hardest thing about their runs is picking your final six. I myself did a Fighting Run recently and it was tough for me to narrow it down.  You have Chesnaught, Lucario, Toxicroak, Hawlucha, and so forth who can give you a great diversity of moves. Water Pokemon can give you a healthy team for role diversity in terms of defense, support, and attack (and Greninja is always nice).  Flying Type also has a lot of gems, most notably Gliscor, Charizard, and Honchkrow. All of these types are available very early and consistently give you new Pokemon to play with as you progress.

The Worst Types

Like I said before, even the “worst” types in the game are still not bad, especially compared to the other games.  But the worst would probably be Rock as it’s available relatively late compared to the other types.  The first pokemon you can catch, Dwebble and Binnacle, are available shortly before you get to the second gym.  Still though, this may be one of the best Rock Type runs in Pokemon due to the sheer abundance and diversity of them.  You neutralize your weaknesses right off the bat thanks to the aforementioned Dwebble and Binnacle! So how about that?  Also, when you arrive to Glittering Cave, shortly afterwards, you can catch a Solrock, Lunatone, Onix, and a Rhyhorn! So you go from drought to deluge in almost an instant!  Also, you can’t go wrong with Tyrantrum.

Dragon, Ice, Ghost, and Fire would be the other subpar types in these games but still, not too bad.  For Dragon Pokemon, you get your first Pokemon, whether a Charmander or Axew, after the first gym and you have some weaknesses to watch out for.  Still, this is a pretty solid team. Hydreigon, Garchomp, potentially Mega-Charizard X and Dragalge, and others give you a nice solid, strong team to fight with!  Fire Pokemon also have weaknesses to Rock and, for Y, Water but at least you have Fennekin as your starter so you can play Fire right from the get go.

Ice Pokemon have all their weaknesses covered which is a shock, but your first Pokemon, Eevee, won’t be acquired until after your first gym.  The same goes for Ghost Pokemon who are also available after the first gym but have their weaknesses covered. On Route 6 you can catch a Nincada and a Honedge at the same time which is nice.

Type Teams

Bug
Ideal Team: Vespiquen/Yanmega, Shedinja, Scolipede, Crustle, Heracross (Y), Durant
Optional: Beedrill, Butterfree, Viviilon, Ninjask, Masquerain, Pinsir (X), Shuckle, Scyther, Wormadam and its various forms, Mothim
First Pokémon: Scatterbug via Route 2 and Weedle (X)/Caterpie (Y) for Route 2 as well but you can catch them both in both versions at Santalune Forest.
Weaknesses Covered?  Regardless of versions, you’re good!

Dark
Ideal Team: Greninja, Pangoro/Scrafty, Malamar, Honchkrow, Houndoom (X)/Bisharp, Drapion/Skuntank
Optional: Crawdaunt, Sharpedo, Absol, Umbreon, Sableye, Liepard (Y)/Mightyena (X), Krookodile, Weavile, Tyranitar (Y), Hydreigon, Zoroark
First Pokémon: Froakie via Starter
Weaknesses Covered? Yes

Dragon
Ideal Team: Goodra, Dragonite, Mega Charizard X (X)/Dragalge (Y), Tyrantrum, Garchomp, Hydreigon
Optional: Flygon, Noivern, Haxorus, Salamence, Altaria, Druddigon, Mega-Ampharos
First Pokémon: Axew via Connecting Cave after the first gym or Charmander in Lumiose City after the first gym for Pokémon X.
Weaknesses Covered? No, X is weak to Dragon.  Y is weak to Dragon and Ice.

Electric
Ideal Team: Heliolisk, Lanturn, Rotom, Magnezone, Mega-Ampharos, Stunfisk
Optional: Manectric (Y), Jolteon, Pachirisu, Raichu, Emolga, Dedenne, Electrode, all the other Rotom forms
First Pokémon: Pikachu via Santalune Forest, before 1st gym
Weaknesses Covered? Yes

Fairy
Ideal Team: Azumarill, Gardevoir, Wigglytuff, Dedenne, Mawile, Granbull
Optional: Aromatisse (Y)/Slurpuff (X), Sylveon, Mr. Mime, Carbink, Klefki, Florges
First Pokémon: Azurill via Route 3 before the first gym
Weaknesses Covered? Yes

Fighting
Ideal Team: Chesnaught, Lucario, Gallade, Pangoro/Scrafty, Toxicroak, Hawlucha
Optional: Medicham, Mienshao, Machoke, Hariyama, Throh (Y)/Sawk (X), Heracross (Y), Poliwrath, Gurdurr
First Pokémon: Chespin via Starter
Weaknesses Covered? Yes

Fire
Ideal Team: Delphox, Talonflame, Charizard, Pyroar, Houndoom (X), Chandelure
Optional: Simisear, Flareon, Magcargo, Torkoal, Heat Rotom
First Pokémon: Fennekin via Starter
Weaknesses Covered? No, in Pokemon X, Rock is not neutralized.  In Pokemon Y, Water and Rock are not neutralized.

Flying
Ideal Team: Talonflame/Charizard, Gyarados, Gliscor, Salamence/Dragonite, Aerodactyl, Honchkrow
Optional: Ninjask, Mothim, Butterfree, Swanna, Pelipper, Swellow, Vivillon, Jumpluff, Swoobat, Crobat, Drifblim, Emolga, Hawlucha, Sigilyph, Staraptor, Mantine, Fan Rotom, Skarmory, Noivern, Pidgeot, Altaria, Scyther, Yanmega, Delibird
First Pokémon: Pidgey, Scatterbug, and Fletchling via Route 2
Weaknesses Covered? Yes, and in more ways than one.  You can have several different team matchups and still have your bases covered.

Ghost
Ideal Team: Shedinja, Aegislash, Rotom, Golurk, Sableye, Chandelure
Optional: Haunter, Phantump/Pumpkaboo, Drifblim, Banette
First Pokémon: Nincada and Honedge via Route 6 after the first gym.
Weaknesses Covered? Yes

Grass
Ideal Team: Chesnaught, Venusaur, Ferrothorn, Exeggutor, Ludicolo, Mow Rotom
Optional: Wormadam, Simisage, Roserade, Gogoat, Vileplume, Bellossom, Jumpluff, Leafeon, Victreebel, Carnivine, Amoonguss, Phantump/Pumpkaboo, Abomasnow
First Pokémon: Chespin via Starter
Weaknesses Covered? Yes

Ground
Ideal Team: Hippowdon, Nidoqueen/Nidoking, Gliscor, Golurk, Steelix, Garchomp
Optional: Diggersby, Marowak, Dugtrio, Rhydon, Quagsire, Graveler, Stunfisk, Whiscash, Mamoswine, Sandslash, Krookodile, Wormadam
First Pokémon: Bunnelby via Route 2, before the first gym
Weaknesses Covered? Yes, and you can even interchange some of these pokemon and still cover weaknesses fine.

Ice
Ideal Team: Aurorus, Frost Rotom, Cloyster (Y)/Lapras, Mamoswine, Jynx, Abomasnow
Optional: Glaceon, Beartic, Avalugg, Weavile, Vanilluxe, Delibird, Cryogonal
First Pokémon:  Eevee via Trade in Camphrier Town after the first gym.
Weaknesses Covered? Yes

Normal
Ideal Team: Snorlax, Pidgeot, Tauros/Miltank, Pyroar, Heliolisk, Wigglytuff
Optional: Diggersby, Linoone, Swellow, Staraptor, Dodrio, Dunsparce, Delcatty, Farfetch’d, Furfrou, Furret, Audino, Smeargle, Kecleon, Exploud, Zangoose, Bibarel, Fearow, Watchog, Chatot, Noctowl, Kangaskhan, Spinda, Ursaring, Lickilicki, Ditto
First Pokémon: Bunnelby, Fletchling, Zigzagoon, and Pidgey via Route 2, before the first gym
Weaknesses Covered? Yes

Poison
Ideal Team: Venusaur, Nidoking/Nidoqueen, Drapion/Skuntank, Scolipede, Toxicroak, Dragalge (Y)/Tentacruel
Optional: Beedrill, Roserade, Vileplume, Swalot, Crobat, Seviper, Haunter, Arbok, Amoonguss, Garbodor, Ariados, Qwilfish
First Pokémon: Weedle via Route 2 in X or Santalune Forest in Y.  Both before the first gym
Weaknesses Covered? Yes

Psychic
Ideal Team: Delphox, Gardevoir, Gallade, Malamar, Slowbro, Sigilyph
Optional: Kadabra, Meowstic, Medicham, Grumpig, Lunatone, Solrock, Swoobat, Espeon, Chimecho, Mr. Mime, Wobbuffet, Reuniclus, Exeggutor, Jynx, Gothitelle, Starmie (X)
First Pokémon: Fennekin via Starter
Weaknesses Covered? Yes

Rock
Ideal Team: Tyrantrum, Barbaracle, Crustle, Aerodactyl, Probopass/Aggron (X), Tyranitar (Y)/Lunatone
Optional: Solrock, Aurorus, Relicanth, Rhydon, Onix, Boldore, Carbink, Corsola, Graveler, Magcargo, Shuckle, Sudowoodo
First Pokémon: Dwebble and Binacle via Rock Smash in Ambrette Town.  This is well into the game but is still before the second gym. Get the Rock Smash TM from the girl outside of the Aquarium.  You can then get Rhyhorn, Onix, Solrock, and Lunatone almost immediately afterwards in Glittering Cave with the fossils following shortly afterwards.
Weaknesses Covered? Yes

Steel
Ideal Team: Lucario, Probopass/Aggron (X), Aegislash, Mawile, Ferrothorn, Durant
Optional: Wormadam, Klefki, Bisharp, Magnezone, Skarmory, Steelix
First Pokémon: Burmy via Route 3, before the first gym.  Don’t worry, you’ll get a Riolu soon after via Route 22, also before the first gym.
Weaknesses Covered? Yes and in more ways than one. Ground is covered by Ferrothorn, Durant, and Skarmory.  Fire by Probopass/Aggron. Fighting by Mawile, Durant, Wormadam, and Skarmory.

Water
Ideal Team: Greninja, Clawitzer (X)/Cloyster (Y), Gyarados, Slowbro, Quagsire, Ludicolo
Optional: Simipour, Bibarel, Crawdaunt, Seaking, Sharpedo, Golduck, Blastoise, Pelipper, Swanna, Wailord, Tentacruel, Starmie (X), Qwilfish, Lapras, Seadra, Relicanth, Vaporeon, Mantine, Octillery, Lanturn, Corsola, Gorebyss, Huntail, Alomomola, Whiscash, Poliwrath, Floatzel, Barbaracle, Azumarill, Wash Rotom
First Pokémon: Froakie via Starter
Weaknesses Covered? Yes, and in more ways than one, you can interchange some of these pokemon for others and still be fine.