How do you go to other regions in Pokemon Omega Ruby, like in Heartgold and Soulsilver?
HearthGold/SoulSilver were a remake of the original Gold/Silver games, also called Gen2.
Thoses games where for the GameBoy Color, and came out at the end of the GBC's life. Whilst nearing the end of G/S's development, the developers discovered a way to compress the game's data enough to add the first game map, and did so.
It was an exception, and was not planned at the start; almost an accident. When making Gen3 (Ruby/Saphirre), they figured they were better off focusing on the new map instead of trying to add one of the previous's game world. Even the scenario made a lot less nod's to the previous game than G/S did to Red/Blue.
When creating the remake, they wanted to remake the game, without making significant changes to the way it was played, and adding such a huge area would be a significant change from the "old game".
TL:DR: You can't in Ruby/Saphirre, the only gen you can do that is Gold/Silver
No, you can't, as Hoenn (the region ORAS takes place in) is too far from the other regions that you can't go to any of them. HGSS (which are remakes of Gold/Silver/Crystal) takes place in Johto, which is right next door to the original Kanto region - that is the only reason lore wise why in that game you could go to another region.