It may be redundant now, but I'm on the NG+ screen and these are the exact options you can carry over:
Character Name Level Jobs Job Level Funds Items (inc. purchased magic) Genome Abilities Norende Village Population Norende Village Status D's Journal Special Moves (Inc. parts/names) Play Time Friends Guests
Nb - scenes in the event viewer carry over automatically, apparently.
A pretty comprehensive list for a NG+ :D
Edit - I'm assuming that ability slots are always reset and are treated as story events? (Don't want to NG+ fully just yet...)
Prinny rule number one: All posts must contain at least one 'dood'. So here it is, dood.