Francesco has already mentioned the "embedding theorem for big line bundles" that you were probably looking for, but I would like to add that this is pretty much the definition of big.
An alternative point one could make is that according to Kodaira's Lemma "big=ample+effective", so a high power of a big divisor is very ample on the complement of an effective divisor.
As for your last question, if $f:Y\to X$ is any proper morphism and $\mathscr L$ is an ample line bundle on $X$, then the image of if $g$ denotes the morphism(!) (say $g$) defined by the global sections of very high powers of $f^*\mathscr L$is , then it follows that $X$. f$ factors through $g$. In particular, then $g$ is an isomorphism to its image (hence an embedding) wherever $f$ is (if $f$ is not birational, then this "wherever" is the empty set).