Much (but not quite all) of what you want is written down clearly in the mostly equivalent language of algebraic groups and homogeneous spaces by Birger Iversen and Robert Fossum. I'll give the journal references, which might not be readily available online. Some of their references are also worth following up.
B.Iversen, The geometry of algebraic groups, Adv. in Math. 20 (1976), 57-85.
[This works mostly just with Borel subgroups and flag varieties, but many of the
ideas are general.]
R. Fossum and B. Iversen, On the Picard groups of algebraic fiber spaces, J. Pure Appl. Algebra 3 (1973), 269-280. [This is fairly general, but they specialize it to varieties $G/P$.]
If you are intrepid enough to try J.C. Jantzen's large and sometimes technical book Representations of Algebraic Groups (2nd ed., Amer. Math. Soc., 2003), he has a careful discussion of ample line bundles in this algebraic framework. But the book is not directed toward algebraic or differential geometry as such. See especially his section II.4.4 for ampleness of homogeneous line bundles on flag varieties, with a remark on the easy adaptation to varieties $G/P$. The criterion here is just that the weight defining the line bundle be strictly dominant (strictly positive at all simple coroots or just positive relative to those simple roots not defining a parabolic subgroup). Jantzen also gives useful references back to work of Kempf. (Ampleness came up in the somewhat related older question on MO here.)