New Game + is unlocked once you've finished the game. It's also unlocked for all lower difficulties than the one you've completed. For example, if you finish the game on 'Hard', you'll not only unlock the trophies for finishing the game on 'Easy' and 'Normal', but also the possibility to play a New Game+ on those difficulties, along with the 'Hard' difficulty.
In New Game+, Joel keeps all his Skills, Skill Points, Training Manuals and Weapon Upgrades. You do, however, still need to find the weapons in the regular order of the game; it'd be way too easy otherwise. You cannot change the difficulty of the game during a New Game+ play through.
If you're serious about getting all trophies, the quickest way to do is by playing through the game on Hard, then on Survivor, and then start a New Game+ on Survivor.
But only if you can finish the game on Survivor without dying or restarting any encounter even once, you may call yourself a true The Last of Us pro.