Let's see, what do I want from Ultra Sun and Moon...
Pokémon following you - This is definitely something I enjoyed back in HG/SS. It's not just for nostalgia's sake (I mean, HG/SS isn't really nostalgic to me anyway)- it makes the world feel a bit more real, and I feel more attached to my Pokémon that way.
Whether it can happen or not on the 3DS is something else- it might or might not make the 3DS chug. I'm sure the Switch can take it and I'd definitely like this mechanic on the Pokémon Switch game, whatever that is, but it might be possible on the 3DS given the data dump we saw months ago.
Move tutors - I've only been in the metagame for a few several months, but yeah, move tutors make a lot of sense. It'd open up a lot for Gen 7's metagame.
Frankly, the Battle Tree area looks like there could be so much more added to it anyway.
Touch screen mechanics from Gen 4/Gen 6 - At first I thought the RotomDex was a bit lame for the touch screen, but I realized how to access the map and Dex from it eventually. It's still not super intuitive, though, and I enjoyed using (some of) the apps from the Pokétch, as well as all the touch screen features from Gen 6 (DexNav, anyone?) Any amount of this would be great for me.
More varied Pokémon - Masuda-san technically touched on this in the Direct, and this might be a given, but I'd like the kinds of Pokémon we can find in Alola to be expanded upon, à la B2W2 (slightly different case, but you get what I mean). Possibly make some Pokémon a little easier to get, but I won't complain if they don't.
Keep the player's emotionless expression - I have a feeling Game Freak sees the player as outside the story looking in, as opposed to some players seeing their character get as involved in the story as any other character. These are 2 philosophies I see in game design a fair amount, and I personally prefer the former. Plus, I find it much more entertaining to see the player with their apathetic face than for them to get engaged in the story. Letting your character have a different face in some situations (such as starting a battle) would be fine with me, though.
Keep the nursery - This is kind of a given, but a must if it isn't. It's very useful for competitive breeding, in my experience. I don't know why they'd do away with it.
Possibly expand upon the inclusion of certain characters? - Remember when we saw Colress on Akala Island, or Grimsley on Ula'Ula Island? It was cool to see them back, but they did nothing except show up in the Battle Tree past that- I was okay with this, but it'd be nice to see these characters get a little more interaction in the main story. This is an "alternate take," after all, so maybe they (and/or other characters) could play a slightly bigger role.
New battle animations? - I had the idea that completely new animations for Pokémon in battle when they were at low health would be awesome and show some more personality of each Pokémon, but that's another 800+ animations to be done. I'd like it, but I won't count on it.
Also, for some reason I got a little disappointed to see no Trainer animation during Mega Evolution this Gen. I assume because Megas basically got relegated to competitive battling so no one would care, but I always liked seeing the animation in Gen 6. This isn't really needed, but I'd like to see it.
As for things I don't want to see...
Version-exclusive clothing - I was at least lucky to get Moon, since I like wearing cool colors over warm colors. Still, I feel like this mechanic wasn't done that well. Either make it easier, or just do away with it. More clothes would be cool, but I'd like this to be dealt with first.
Lillie's character not being constantly shoved down my throat - Look, I like Lillie's character enough, but I felt like the game was constantly shouting at me to love her character, when I just didn't care that much for her. Especially near the end, Lillie's personality felt a bit forced to the point it kind of hurt. I'm probably in the minority here, but I'd like them to change what happens with Lillie so that maybe people like me can enjoy her more- I'd be down for her being a Trainer in these sequels from the very beginning.
Defending your title - I didn't like this at first. It felt wrong to not have an actual Champion to challenge despite battling Prof. Kukui at the end.
I came to like this idea more when I saw that other characters challenge you (that Youngster? Seriously?), but I still prefer things as they were done in previous generations.
Some things I'm not sure of...
Gyms/Trials? - Some people want Gyms back, others want Trials to stay. I'd like both if we can somehow do that, but if it has to be one or the other, I'm not sure.
If Trials came back but were totally revamped, and we got a new Trial, I'd be happy with them. Otherwise, I'd be happy seeing Gyms come back- it'd make sense given those empty spaces in Alola.
Festival Plaza? - I don't know what it is about the Plaza, but I hear from a lot of people that don't really like it.
Personally, I like the Plaza as it is, but I can see some improvements they should make- for one, I think it'd be smart if you were able to just upgrade shops instead of hope the next shop you can get is just a better version. Whatever they can do to make the Plaza better, I'll be down with.