Glad you asked!
A great post-game quest, I think this would be strongly suggested by me to try out. Without spoiling, I can say you get a new Mega and also two legends. Here is a step-by-step guide, but it does have spoilers.
I think this is an unique feature of the game that might seem a bit boring, but is rewarding too. To start doing these, approach any contest spectacular building. They're the fancy ones in Lilycove, Slateport, Fallarbor and more areas. After beating your first contest, you'll get a nifty contest pikachu. You can also get cool items from fans when you beat high-ranked contests, and a special gift when you beat master rank in all contests. Beating all master ranked contests with one Pokemon also gets them a sparkling effect when they enter battle.
Make a cool secret base
Get some decorations and make the best secret base ever! While it is a little limited, you could make:
-A small house
-Your own gym
-A display of dolls
And more! You can also get secret pals, and some of them can even get you master balls. I think this would also be a nice thing to do.
There are many more things to do, like (as you stated) catching all the Pokemon, breeding/chaining shinies, EV train and breed to get the strongest Pokemon anyone could find, etc. The battle resort and wifi-battling might be interesting too, or you can just keep re-battling the elite four. Theres also restarting the game to do a nuzlocke or just to play though it again.
Hope I helped :D!