The idea is to play through to the 32-cube ending, which should present itself as a natural progression. Once you finish the game, you have no choice but to start a new game (the New Game+) if you're so inclined to complete every puzzle (or "209.4%" the game). The NG+ is the same as the first playthrough (everything carrries over) except for one bonus feature, which only actually helps with one puzzle-type:
The white and purple pillars (not the U-shaped tuning fork pillars) are the only puzzles that depend on the first-person vision New Game+ bonus.
The rest can be solved without it in the first playthrough. There is no in-game method for telling which puzzles have all of its pieces. It requires unusual attention to hidden/obscure codes and nuances. One could contend that the game's main conceit is withholding information from the player (see the monolith debacle). Later (optional) Easter eggs include finding the 64-cube door and collecting the heart cube pieces. You will most likely check the Internet for answers before exhausting every in-game clue, and, presumably, Fez was designed to be played with that in mind.