I would like to use your environment to your advantage, giving fire types, Grass types and flying types an advantage on a sunny, clear day for example. I'd also love to be able to actually level up my trainer, giving them more knowledge as they goes along will allow them to make better use of their Pokemon's abilities.
The option to join teams would also help, similar to the anime you are allowed to have followers to assist in regular battles and offer support in gym battles. Your followers would be able to level up as well and you should add incentives for teaming up either by accessing unique abilities or allow you to gain trust with them which would help even more in battle. Solo players would get a number of options but those connected would even allow them to team up for a time.
Those things are just suggestions and I'd be happy with or without them but things that do genuinely bug me are sound and saves. First off I'd like to say that the old 8-bit sounds do have their charm but its starting to wear thin, hell even the GBC was able to have voices why does the DS still have to use the same sounds as a game boy?
Second of all, For Fucks Sake WHY can't I have more than one fucking save per cartridge?!? I know some people don't mind it but I'd rather be able to let my friends and family borrow my Pokemon without flushing 150 hours down the toilet!