So I'm playing through the game atm, currently at chapter 6, just wondering, is there any kind of New Game + in FFXIII?

Nope. You unlock one or two things after defeating the final boss, and it saves but you're back at the save spot before facing off with the final boss. At that point you're free to go back and complete missions and such. But, alas, no New Game +

Not a New Game + but after you beat the final boss you unlock the 10th level of the crystalium and you can go back and finish the side quests.

No. However after you finish the game the last tier of the crystarium unlocks. In the chapter 13 area you'll be able to go back to the chap 11 and 12 locations, but it doesn't let you back to anywhere that you now. Once it saves you'll be able to load up the game just before the boss and you have the chance to finish up missions and do the harder sidequests, getting the rest of the achivements/trophies.

There is a ton of stuff todo after the final credits have rolled. You will carry on having fun for hours after so don't worry.