The colon has a number of uses, one of which is to elaborate on the previous statement, such as this from "The Devil's Dictionary" by Ambrose Bierce:
CRITIC, n. : A person who boasts himself hard to please because nobody tries to please him.
In this case, yes, you can say what follows the colon is a noun phrase (though I'm not sure why that matters). In other cases, the elaboration can be a complete sentence:
The most important message of the film is this: You can't ever really go home again.
The semicolon is different; it is used to join together two complete sentences that could otherwise be separated by a period (as in this sentence). However, in creative writing the semicolon is sometimes used as an alternative to the comma, as in this quote from "Cat's Cradle" by Kurt Vonnegut:
Only I was going to kindergarten; Frank was going to junior high; and Father was going to work on the atom bomb.