Bring back Contests! Not Super Contests, but the original R/S Contests, with halls across the entire region that you can go to while you do your badge hunt. That was always a ton of fun for me, raising one team for battles and another for Contests, playing through with them simultaneously (which is why I hate how Emerald combined them all into one hall). To me, Super Contests were a step in the wrong direction by adding an action sequence, and Pokemon Musical is the absolute worst thing to ever come into the franchise.
Besides that, I'd like to see the removal of HMs. Instead, I like the idea of "if a Pokemon could conceivably do this, it shall be able to." I hate using HM Slaves, and as such never do, but the downside to this is that the Water Pokemon on my team is stuck with Surf, Waterfall, Dive, and only a single move of my choice. I think just getting the qualifying badge should be enough, instead of having to waste a move slot to fly on my Pidgeot or have my Scyther cut down a shrub. By removing HMs, this would also help with some "things that just don't make sense," like being able to fly on a Spearow or surf on a Mudkip, by allowing the Pokemon to use these moves but disallowing the usage of these Pokemon to traverse obstacles.
I also love that Acid Rain idea that was posted earlier.