After the Delta Episode, head downstairs from your room to meet your parents once again. You'll find that Mr. Briney has mailed you an S.S. Ticket, which will let you ride the ferries in Slateport and Lilycove to the Battle Resort. You've probably earned a bit of relaxation time having saved the world twice now.
Before leaving, though, be sure to speak with your mother to obtain some Latiasite in Omega Ruby and some Latiosite in Alpha Sapphire; this is the Mega Stone for the Lati opposite your version, we guess in the intent of the Eon Ticket event or in the case you trade or import one.
As you leave your home you'll hear a familiar scream. Quick, run to Route 101 like in the start of the game! Poor Professor Birch is in a spot of bother and he needs your help once again.
Examine his bag and you'll be able to use one of the Johto-region starter Pokémon to help him out of his plight. At this point, it doesn't really matter what you pick, so pick whichever you like most. No battle, though, turns out the Shroomish is friendly...
Later, if you come back to this route again from your home in Littleroot, the Professor will request your assistance two more times, allowing you to obtain an Unova starter Pokémon (after the Delta Episode) and a Sinnoh starter Pokémon (after rematching the Elite Four).
Johto starters (can only pick one)
Unova starters (can only pick one)
Sinnoh starters (can only pick one)