To unlock New Game+ (for another play through the game), complete the entire game and save thereafter. Then, load your data. You'll restart the game, but have all of your old gear, statistics, and more.

Here's a great way to level-up once you've already beaten the game. Head back to Tarturus, where you can find a door called Monad. The monsters in Monad are stronger than most other monsters in the game, but your main character should be able to take them out pretty easily. Spending time there can level-up your weaker characters in only an hour or two.