Yes, you get to higher realms through training...lots of training.
If I am correct, the rule is realm 2 = lvl 20; realm 3 = lvl 30. (Is that correct, somebody who knows for sure?)
As for regeneration, the only way to make that faster is take a few mana potions.
As for having more mana points in the first place, its intelligence and will: charisma helps a bit as well I think.