To summarize, once you've switched to the PS4 game, Square Enix says there's no playing again on PS3 with your existing copy, so be sure it's what you want to do. If you'd prefer to play on both systems, you can purchase a PS4 version separately.
As for the how, the PlayStation Blog has a step-by-step guide to take you from PS3 to PS4. Once you're there, you'll be able to explore among PS3 and PC players, and you can use your characters across all platforms that you've activated with your account.
Final Fantasy XIV: A Realm Reborn