
This is how the various types break down for each team, except Fairy replaces Bug in the Air Nomad team, and Bug is free use for all nations like the Normal Type.
In Pokemon Sun, I'm using the Air Nomad team first, so I can only use Pokemon that are some combination of Flying, Psychic, Ghost, Fairy, Bug and Normal. I can get them either in game or through Wonder Trade.
All those that qualify must remain on the team or in the box, including those that can qualify through a later evolution, except duplicates.
If a Pokemon Qualifies, but its evolution does not, it may not evolve to continue to qualify for the team.
Mega Evolutions of different types can be used, because its a temporary transformation and not a permanent change.

Book 1: Air
And starting the journey of the Air Nomads, I of course name the character Aang.

To begin, I started with Rowlet, but because its part grass and because the Starter Team had all Gen VII starters, I'm not going to use them for any of the Avatar Teams. But very quickly, I caught a Pikipek, which will serve as the Starter Pokemon for this Team.
By the time I got through the Iki Town celebration and started at the Pokemon School, I had added a Drowzee and an Abra.
Abra, of course only knows Teleport, so it won't be too useful in combat just yet. However, Pikipek and Drowzee are more than enough for the students at the trainers school.
Losers
Along the way, I picked up a Chansey and a Caterpie that I put in the box. Moving in through Hau'oli City, meet our good buddies Thing One and Thing Two!
Their shirts aren't numbered, so I
don't know which is which
Pikipek with Rock Smash is more than enough for Ilima's little test, and then its off to Route 2 and the Verdant Cavern!
In the cemetary, I caught a Misdreavus to add to the team.
Along the way I traded for a Cutiefly and a Flabebe, which I added to the team. I also traded for a Ledyba, Mimikyu, and Komala and caught a Spearow, all of which are currently in the box.
Moving on to the berry farm, Thing One and Thing Two return!
Misdreavus takes care of his Drowzee, and we move on to the actual reason we came to Route 2, the Verdant Cavern!
In the cavern, both Pikipek and Abra evolve, and Trumbeak with its Rock Smash gets us through the cave and to the Totem Gumshoos!
Aaaahhh!!!
However, Trumbeak can't beat both Gumshoos and the summoned Yungoose, and after defeating Gumshoos, loses to Yungoose. Luckily, Kadabra had the speed to take it out in one shot!
One Trial down, 7 to go! Plus 4 Grand Trials!