There's a lot to gush about in Sun/Moon so I'll try (and fail) to be brief):
1. The fact that somebody at Gamefreak took a look at my Pokemon wishlist when designing the new 'mons. The big ones being:
-Poison finally got there first Legendary Pokemon in Nihilego (along with some great new additions in Toxapex, Salazzle, and even Alola Muk)
-Bug got a bunch of great additions picking up where Gen 5 left off (Araquanid, Golisopod, Pheromosa, Vikavolt, and even Ribombee, along with Lunge and the Leech Life buff)
-Rock got many new breakouts from the "slow tank" mold (Lycanroc, Minior, and finally a true special attacker in Nihilego and Diancie retroactively due to Power Gem)
-Ice finally got some much needed buffs in Slush Rush and Aurora Veil, along with more Snow Warning users.
-Terrains became much more usable (arguably too usable...)
-Only one new legendary dragon and he's more a space lobster than anything.
Really, about the only thing I want now is the dolphin pokemon they've been avoiding for years.
2. Boss fights and challenge, sweet mercy I missed you. And it's not just the really great Totem Pokemon battles either; I love the way that most of the rivals and key trainer fights have some strategy to them. Guza shows off priority with First Impression and Sucker Punch, Kukui abuses Stealth Rock, even Nanu uses a confusion and flinch element to him. And the introduction of Z-trainers to show off specific strategies goes pretty far in closing the in-game vs post-game gap that's been a problem since the inception of the series. I love how when you get Substitute they game tells you the right ways to use it (status protection). I've always wanted the metagame to be accesible to everyone and they've finally started to address that in the main-game!
3. Dark finally got a gym leader. Well, "Kahuna" but the spirit is there. Love you Nanu.
4. The characters are great but everyone above has mentioned them so I don't have much more to say.
5. I love how you can tell where all the trial captains live without anybody actually telling you or any sign labeling it. I also love how you find their rooms often a long time before you meet the people that lived there; the amount of story told by seeing Guzma's room loooong before you meet the guy was superb. It's Gamefreak capitalizing on the atmospheric and exploratory storytelling that Pokemon is best suited for.
6. The world. The biggest one for me, this is the first time I really felt that Gamefreak created a world inhabited by people and Pokemon. Before they always seemed separated slightly, like the mechanics, story, and setting had pokemon in it but was largely just full of humans first and foremost. HG/SS came close to touching it, but still interactions between pokemon and people felt artificially limited.
These games did a lot to bridge that. The Machamp at the graveyard, the boy and his Drifloon, the little way that Lillie or Kukui always mention whatever pokemon is in the first slot, the way your starter chooses you, the Oranguru with a master in jail, the Eevee sidequest, even the way Meowth wakes you up, all these tiny details finally made the game feel like a living breathing real place and less like a video game.