Concerned about whether my diamond is real?

The rainbow colors you speak of is called the "fire" of the diamond compared to the "brilliance" of the diamond which is refracted white light sparkle. The stone should have come with papers. See if you can do this even though it's mounted already. Lay it on some newsprint and try to read the type, in a real gem the type will be unreadable in a simulated gem in will be readable. Yes, take it in to be appraised, it may still have value even if not a genuine diamond.