I would suggest you two references; the first one is Demailly's survey on Hodge theory ([B3] here: http://www-fourier.ujf-grenoble.fr/~demailly/books.html ); exercise 15.11 explains exactly how to construct sections with desired vanishing order at some point.
This is a consequence of Nadel's theorem applied to singular metrics on ample line bundles; the idea is to impose the metric to have a logarithmic pole at your point, so that Skoda's lemma (the easy part) ensures you that any section of L twisted by the multiplier ideal of the metric has to vanish enough along P. As a consequence, you can deduce Kodaira's embedding theorem.
The second reference is the one already mentionned by Donu Arapura. More precisely, there is a lemma in Lazarsfeld' PAG I (I will check the exact reference)edit: Proposition 1.1.31 p.23). It shows how elementary linear algebra (+ RR) can give a lower bound on the vanishing order of sections some section of a (say nef and big) line bundle with high big enough top intersection.