There's a lot for you to do after you've beaten the game, but you can do it in pretty much whatever order you'd like. Below is a list of some of the things that happen after you fire up the game once you've beaten it to give you an idea of what you'd like to do now.
You wake up in your room. When you go downstairs, it'll look like you have two moms down there at first, but one of them is just Looker (yeah, the guy from Platinum). He'll say how he's looking for the Plasma sage dudes and will give you the Super Rod.
When you leave your house, Cheren and Bianca will be outside. Cedric Juniper will come over and upgrade your Pokedex. This is when you get the National Dex! You don't have to do anything special to get it, don't worry.
From here, you can search for the six remaining Plasma sages across Unova.
From here on in, all of the Trainers you're going to fight use Pokemon at level 60-65! No joke! There's nowhere in particular you can go to level up, except fighting against them and other wild Pokemon at level 48-55 (or up to 60+ in the thick grass!). However, don't let their levels intimidate you, because they usually aren't tougher than the Elite Four, N, or Ghetsis were due to their lack of powerful moves or high stats.
You can go to Opelucid City and then head east to go towards the Giant Chasm, where you can eventually catch the legendary Ice/Dragon-type Pokemon, Kyurem L75.
You can go east of Nimbasa City and across the bridge to reach either the White Forest or the Black City, depending on your version. You need to interact with people in the Entralink in order for stuff to happen here.
When you cross that bridge, the Shadow Triad will give you the Lustrous Orb, Adamant Orb, and Griseous Orb.
The Abundant Shrine is around White Forest or Black City, up some waterfalls in the route to the northeast. You can take both Tornadus and Thundurus there to encounter Landorus L70.
You can go back to the Relic Castle and, after some exploring, you can find an Volcarona L70 down there that you can catch!
Undella Town on the far east has a girl outside that will give you HM06 (Dive) if you talk to her, which can be used to dive off the coast of the town. You can get most of the Plates for Arceus down there, as well as a ton of loot that can be sold for millions!
Cynthia is available for you to battle in a house in Undella Town. Beware, because she uses Pokemon at level 75+!
The Poke Transfer Lab just to the west of Black City or White Forest can be used to "migrate" Pokemon from Diamond, Pearl, Platinum, HeartGold, or SoulSilver if you have two Nintendo DS systems available to you. There are no badge requirements on any of the games you're transferring Pokemon from, and the transfer is one-way.
The Elite Four are now far, far stronger, possessing tough level 71+ Pokemon on their team. If you beat all of them, you'll be able to take on the Champion, too, who uses Pokemon from level 75 to 77!
You can take on Game Freak Morimoto in the Game Freak building in Castelia City. This trainer is modeled after the programmer, game designer, and even Pokemon designer himself, Shigeki Morimoto. He has a full team of Pokemon with levels 75 to 77, but is easier than Cynthia or Champion Alder.
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