This is the team I used in my most recent Sun playthrough. Its got a Sun exclusive, and I don't know what else could really fit in that slot from Moon, so this will unfortunately be a Sun exclusive team. I usually build my in-game teams by starting with a core, usually fire/water/grass, and going from there. I managed to fit 3 different cores in this one: Fire/Water/Grass, Dark/Psychic/Fighting, and Fairy/Steel/Dragon.
I have listed these in the order they're avaliable in the game, so you shouldn't have any trouble with backtracking. That being said, there's only one Pokemon that isn't available on the first island, so you shouldn't have any problem putting this team together.
For starter, I chose Litten. I picked Litten because it was the only starter I hadn't used yet.
The name is ultimately up to you, if you want to give it a nickname at all, but I had been watching Naruto recently, so I named mine "Killer Bee" since he gets Darkest Lariat.
For moveset, I went with these:
-Flare Blitz : Strongest Fire STAB he learns
-Flame Charge : A lot of people use Swords Dance, but I thought Incineroar hit hard enough, and needed more speed instead.
-Darkest Lariat : His signature move, and Dark STAB
-Leech Life : For recovery and coverage
For Item, I went with Incinium-Z
Next is Pichu. Its a huge pain to train up, but once it evolves into Alolan Raichu (which could take a while If you want Thunderbolt on it pre-E4) it becomes pretty good. Originally I had an Espeon, but when I realized I didn't have hardly any coverage for flying types, I went back and got Raichu instead, since it kept the psychic type I needed for my core, and gave me electric type to cover my flying weakness.
I named mine Pancakes, after Joey and Emvee's Raichu, if you follow the MBL.
For moveset:
-Thunderbolt/Thunder : A pain to get, but is worth it, as its the best STAB move it gets that doesn't miss. You could use Thunder, though, if you want to switch in to Raichu after leading with the next team member.
-Psychic : One of the only psychic moves it gets, its STAB, and hits hard.
-Volt Switch/Electric Terrain : Volt switch is to get you out of situations you don't want to be in, against anything except a ground type, but, If you're willing to go through the effort, and you don't want Raichu to have a practically useless ability, you can catch a Togedemaru, get electric terrain on it, and breed it onto a Pichu. Just like thunderbolt, it's a pain to get, but is worth it. Electric terrain is a good buffer for your electric type moves (50% increase) and it allows your ability to not be useless. Up to you which one you choose. Both are good.
-Nasty Plot : I thought Raichu was plenty fast, so I kept it as a Pichu long enough to get Nasty Plot. Good setup move, helps Raichu to hit even harder.
For Item, I went with Aloraichium-Z, but if you decide to forego Thunderbolt, you can't use Stoked Sparksurfer, so you could go with something like a magnet, or the electric/psychic Z crystal.
Next, Is a Pokemon I'm honestly surprised I didn't see more of in this thread, and is basically the reason I decided to make an answer to this thread. Its Wingull! Pelipper is honestly a complete monster in this game, so long as you make sure your wingull has Hydration as it's ability. If it does, It'll get Drizzle when it evolves, and It makes Pelipper incredibly strong for In-game purposes.
I named my Pelipper Stormbringer, because that's literally what he does. Generic? yes. Fitting? Also yes. Name it whatever you like.
For Moveset:
-Hurricane :
Normally I wouldn't use Hurricane for story stuff. It can miss a lot, and you really don't need it most of the time. But, with Drizzle setting up the rain, Hurricane becomes 100% accurate, so there is no drawback to using this move. Its extremely strong Flying STAB that can't miss as long as the rain is up.
-Scald/Surf :
Depending on what you prefer, the utility of the burn chance of scald, or the higher power of surf, both suffice as good water STAB for Pelipper. I used Scald, because if you need that extra power, I had Pelipper hold Waterium-Z. Nothing survives a rain boosted STAB Hydro Vortex. Nothing.
-Ice Beam : You get this really late in the game, but it provides nice coverage at about the time you need it. If Raichu dies to Kahili, you'll have some bonus coverage for her in Pelipper with Ice beam.
-Roost : Recovery. Pelipper is pretty bulky naturally, so having recovery is great.
For Item, as I mentioned above, I chose Waterium-Z. I don't think a single Pokemon I used it against survived that Hydro Vortex. But, if you prefer, you could get the damp rock in Malie city, and use that to extend the duration of your rain.
Since the introduction of Fairy types (I think fairy's are broken tbh.) I've included two things on my team: A Dragonslayer (Fairy type) and a Fairy Killer. The next Pokemon is Alolan Diglett, my Fairy Killer. In my opinion, there's only two bad things about Alolan Diglett. One: It's horrendous HP stat of 35, and Two: I hate it's design. Two is a nitpick, though, which is why I used him!
I named mine Loreal, because he's worth it.
For moveset:
-Iron Head : Good STAB. Makes sure any fairies bite the dust.
-Earthquake : He learns it kinda late, but Earthquake is powerful. If you have a ground type, you use Earthquake.
-Stone Edge/Rock Slide : These are for Coverage. For some reason, rock type moves just do not like to have perfect accuracy. So, either one is fine. I chose Stone Edge for mine personally, since it's got higher base power for Continental Crush if you want to use it, but, if you prefer more accuracy, go with rock slide.
-Sandstorm : This is utility. Provides chip damage, and if you use Z-Sandstorm, it buffs Dugtrio's speed. Mostly its here because I don't think Tangling Hair is that great of an ability, considering that Dugtrio has that terrible HP of 35, so Mine had sand veil, and I put this on here to take advantage of that ability. It helps Dugtrio stay alive more often, since he has evasion. That's his way of taking damage, is to avoid it altogether.
For Item, I used Rockium-Z. If you want some extra speed, you can use Z-Sandstorm to both get your speed boost, and setup for Sand Veil. It also gives you options if you need that strong rock move in Continental Crush.
Now that we have a Fairy Killer, I needed a Dragonslayer, and I chose Cottonee. Cottonee was a tough nut to crack. I honestly wasn't sure how I felt about her untill she finally evolved into a Whimsicott. Arguably, this was the hardest Pokemon to train on the whole team, because on one hand, it's attacks only tickled the enemy, and on the other hand, Prankster Leech Seed is extremely good at stalling. Idk, it fit two of my 3 cores. It finished off my Fire/Water/Grass core, and built upon the Fairy/Steel/Dragon core. It fits well, but definitely was pretty mediocre for most of the game.
I named mine Kena, because it was a female and I like that name a lot.
For moveset:
-Giga Drain : Decent Stab, but the big part is that it provides recovery.
-Moonblast : A dragonslayer needs a sword, and this move is any fairy type's sword.
-Hurricane : This is here for coverage, and because it works together with Pelipper's ability. Normally I wouldn't use this, but Whimsicott doesn't get much in terms of coverage, so, it's here.
-Leech Seed : Leech seed saved this Pokemon's life so many times it's unreal. Prankster made sure I didn't get flinched before setting up the leech seed, and between this, and mega drain, Totem Wishiwashi slowly but surely faded away.
For Item, I had already used a lot of Z moves, and while you could use yet another Z move on it, I chose Big Root. Big Root increases HP recovery of both Leech Seed and Giga Drain, it just fit so well, that I didn't think a Z-move would even be better in this case.
Finally, it'll be a loooong time before you pick up the last member of the team, but I think the team is more than capable of holding it's own untill Kommo-o finally becomes available. I chose Kommo-o because he finished off the other two cores I had been building. Fairy/Steel/Dragon, and Dark/Psychic/Fighting.
I named mine Susanoo-o as a play on his name, and, like I said, I'd been watching a lot of Naruto. Haha.
For moveset:
-Dragon Claw/Clanging Scales : While Clanging scales is powerful, I went with Dragon Claw because Clanging Scales only has 5pp. If you've been saving your PPups, you could use Clanging Scales if you prefer, but I had already used all of mine by that point, so I kept Dragon Claw.
-Brick Break : I'm honestly surprised that Kommo-o doesn't get harder hitting fighting type moves. Yeah, he does get Focus Blast, but that's low accuracy and low pp, so I opted out of that for the same reason I opted out of Clanging Scales. Brick Break is 100% accurate, decent base power, and has the added bonus of breaking through light screen and reflect, if anything sets one of those up. I hope he gets Close Combat in Ultra Sun and Ultra Moon. It'd be so much better. Drain punch would be really nice, too.
-Poison Jab : Coverage. Although I wouldn't dare put Kommo-o in the face of a fairy JUST to poison jab it, if it comes down to it, and you need it, he can hit fairies hard with this.
-Bulk Up : This is mainly here because I like to have some form of status or setup move on every Pokemon, because there's always the chance you might need it. Its better to have it and not need it, than to need it and not have it.
For Item, I went with Fightinium-Z so that I could have that high base power fighting move if I needed it.
Finally, if there is one piece of advice I could give you for this team, it would be to never underestimate Pelipper. He was the MVP without a single doubt in my mind. He destroyed absolutely everything. Hope you give this team a try, and enjoy it!