The different endings aren't difficult to get. All you have to do is finish the game at different points in the game (I know that sounds odd) by playing New Game +. Winning in different time periods SLIGHTLY changes the ending, as well.

For example, when you complete the Black Omen and defeat Lavos in the Present, Lucca makes a passing comment that R-66Y may not exist in the new future. If you finish the game in the future, since you didn't alter history before that point in time, then that future was still the same. Thus, R-66Y still existed.