Surely you need some assumptions on your variety, for example for every $n$ there is a smooth surface of degree $n$ in $CP^3$ that contains a line, for $n>3$ such surfaces are minimal and of general type.

For hypersufaces in $CP^n$ the most optimistically you can hope that a generic hypersurface of degree $2n+1$ is hyperbolic (hypersurfaces of lower degree generically contain lines I think). This is related to a conjecture of Kobayshi, you can check here http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/hyp_generic.pdf

For surfaces in $CP^3$ the minimal degree for today that is known is $21$. Recently there was a real progress in proving of Kobayashi conjecture http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.2346v4.pdf though the obtained bound is superexponential in $n$

I remember ther was a very nice review of Clair Vosin on this topic but I could not find it now on the internet

Surely you need some assumptions on your variety, for example for every $n$ there is a smooth surface of degree $n$ in $CP^3$ that contains a line, for $n>3$ such surfaces are minimal and of general type. So for large classes of varieties of general type you need an additional assumption that the variety should be generic.

For hypersufaces in $CP^n$ the most optimistically you can hope that a generic hypersurface of degree $2n+1$ is hyperbolic (hypersurfaces of degree $2n-1$ and less always contain lines). This is related to a conjecture of Kobayshi. There is a very nice review of Claire Vosin on different aspects of hyperbolicity of complex projective manifold that you can find here http://people.math.jussieu.fr/~voisin/Articlesweb/harvard.pdf (this also contains the result mentioned by Tony Pantev)

Recently there was a genuing progress in proving of Kobayashi conjecture http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.2346v4.pdf though the obtained bound is very far fron optimal, worse than a triple exponent of n. At the conference due to 80 birthday of Atiyah Kirwan annonced that she can get a much more realistic bound.