Contests: I would like to see contests come back. I loved the Ruby/Sapphire version of contests... even if I wasn't great at them. Ruby is the only game I've managed to get a Milotic, and it won me all of the Beauty contests. Raising a contest attribute, instead of just EVs, makes Pokemon interaction even more rewarding.
Starter: A throwback to Pokemon Yellow. Start us off with a Fairy pokemon, instead of the three starter pokemon. Maybe even have that Fairy starter be a new/unique Pokemon.
Racing: I keep seeing how Rhyhorn and Skiddo are used for racing, but we never get to in the game? Booooo!
Best Friends: Let us have a Pokemon follow behind us while adventuring. Maybe not a Wailord (seriously? how do they even have that in the Battle Mansion haha), but everyone needs a rebelious "Pikachu" that refuses to get in their ball.
Pictures: I don't want to see just a picture of my character in the game, I want my Pokemon to pose with me for pictures as well. It would take a lot of programming, and might skew what pictures end up looking like, but the picture-taking at the moment is easy to get bored with.
GTS: We need to be able to see the natures on Pokemon that are being offered for trade. It's important! Also, a clear indication of what original region the Pokemon is from.
Everyone's saying that one. Maybe one use items that serve HM's? Like you could buy axe's from the store, and each axe cuts one tree, or a Surf Board like you get a Bike. Things like that.
Of course, simply making them removable easily could help a lot.
Or be able to use move equivelants in place of them. Why can't a Slash chop down a tree? Flying/Levitate or Psychic types should be able to carry you once they hit an evolution or two. Same with Water types being able to drag you along the water.
At least they've fixed Rock Smash and Flash for the time being.