What can I do after the Delta Episode?

This is what you can do in the post-game:

Post Elite-Four

• Collect Starters from 3 different regions from Birch. (Exit your house, and he'll call for help. You can do this 3 times and get starters from each of these regions in this order: Johto, Unova, Sinnoh)

• Collect an Egg from the Old Woman near the hot springs in Lavaridge Town that will hatch into Togepi

• Use Mirage Spots to capture legendaries (see this page for more details where to find them)

• Battle Resort is open, Move Tutor is available, Battle Maison is open

  • Once you have beat a Battle Chatelaine in the Battle Maison, you will be able to rematch Wally once a day

• Complete Hoenn Dex and get Oval Charm

• Complete National Dex and get Shiny Charm

Post Delta Episode

• Collect 1 Beldum from Steven's house in Mossdeep City

• Collect 1 Sharpedo and 1 Camerupt from 2 grunts inside the Battle Resort

• Collect the remaining Megastones. Here's where to find each one:

  • Audinite - Battle Resort (Talk to Looker at the Battle Resort)
  • Blastoisinite - S.S. Tidal - Find it on the ground.
  • Cameruptite - Team Magma's Hideout [Ruby] / Battle Resort [Sapphire] (Get it during the Delta Episode [Ruby] or get it from a Team Magma Grunt at the Battle Resort [Sapphire]).
  • Galladite - Fallarbor Town - Talk to Professor Cozmo in his lab.
  • Gengarite - Battle Resort - Find it on the ground.
  • Latiasite - Litteroot Town [Ruby] - Talk to your Mom in your house.
  • Latiosite - Litteroot Town [Sapphire] - Talk to your Mom in your house.
  • Metagrossite - Pokemon League - Defeat Steven once again after the Delta Episode.
  • Salamencite - Meteor Falls - Talk to the old woman there after the Delta Episode.
  • Sharpendonite - Battle Resort [Ruby] / Team Aqua's Hideout [Sapphire] (Get it from a Team Aqua Grunt at the Battle Resort [Ruby] or get it during the Delta Episode [Sapphire])

Source Reddit.

After completely beating the game, I personally like to try remaking my team and challenging the elite four again. You can also collect all of mega stones, as mentioned prior. Another fun attempt is to try and catch all the legendary Pokemon, and it's also fun to wonder trade repeatedly until you get something decent. These are all just opinion though. If all else fails get a friend to come play with you, or trade your favorite Pokemon to them for safe keeping and start over.

You could just hunt for shinies, legendaries, or complete the Pokedex, every Pokemon maniac's dream, because you know, in Pokemon X & Y there was the Pokemon maniac, the one with a white shirt and posed with Pokeballs in between his fingers.

Once you finish ORAS Delta Episode I would finish the Pokédex and if you have Pokemon X or Y finish that the same way you did with ORAS (omega, ruby, alpha sapphire).