Now quite common in RPGs and even the wider gaming community as a whole, the concept of the New Game+ was still pretty novel back in 1995; in fact, Chrono Trigger was the first game to even use the now-well-known term of "New Game+." Of course, it wasn't the first game to offer such a mode, but if you ask around to your gamer buddies, don't be surprised if CT is the first name that comes to mind when related to the concept.
Not too hard. Beat the game however you see fit (most people on their first playthrough beat the game after completing the Black Omen, though any way will do), watch the ending you're given, reset, and select New Game+ on the menu. You'll then pick a savegame to use; stats, equipment, etc. will be carried over from that savegame into your new game (plus!).
Virtually all of your item inventory comes over, as well as your characters' statistics. The characters themselves will not, of course, as the plot of the game does not allow you to have a full roster of characters at the start. Similarly, some items and equipment that are important to the plot of the game will not carry over, and you will have to find them again as part of the gameplay process. Finally, you'll again be broke when you start your NG+; of course, that shouldn't matter too much as the inventory and stats that carry over will ensure that you don't need that much money for a while.
Since there are many more things that do than don't, it's much easier to give a list of what does not come over.