As always, the source you use may be related to what your goals are. To give some perspective, recall there are several ways to define sheaf cohomology, and Serre and Hartshorne feature different methods. Serre used Cech cohomology, and there the important long exact sequence property does not always hold. He was able to prove however that it does hold for "coherent" sheaves. One big advantage of Cech theory is its easier computability in specific cases, such as on projective space. Hartshorne presents first Grothendieck's theory of derived functor cohomology, but then proves it agrees with Cech cohomology before using that theory to compute the cohomology of coherent sheaves on projective space. But if you want to learn the derived functor theory you must choose Hartshorne over Serre.
The distinction made above between schemes and varieties is also relevant. Serre teaches Cech cohomology on varieties,and Hartshorne presents derived functor cohomology on schemes. If you are only interested in varieties, or prefer learning cohomology in the easier setting of varieties, then you may prefer Serre's FAC. Another good source is the book Algebraic Varieties by George Kempf, where the derived functor theory is presented on varieties and used for basic computations, including coherent cohomology of projective space and even the full Riemann Roch theorem, before being linked with Cech theory. So if you want to learn to make computations with the abstract derived functor theory you might prefer George's treatment, although some details are missing there, and some misprints exist.
Finally there are slight differences in Serre's and Hartshorne's results which can be relevant in some settings. E.g. in Beauville's book on surfaces, he uses Cech theory to relate rank two vector bundles on curves with ruled surfaces. To prove that all ruled surfaces arise from vector bundles he then uses Serre's result that Cech H^2 vanishes on a curve with coefficients in any sheaf coherent or not. (He also gives a second argument.) But this vanishing theorem for Cech cohomology does not follow from Hartshorne's treatment, since he proves vanishing for derived functor cohomology but does not relate derived functor and Cech cohomology on non coherent sheaves above degree one.
There is a sentence in Hartshorne, at the end of chapter III, section 2, page 212 in the 1977 edition, which says that Serre proved vanishing for "coherent sheaves on algebraic curves and projective algebraic varieties", whereas the correct statement would be that he proved it "for curves, and for coherent sheaves on projective algebraic varieties". Since Robin is very careful, one wonders whether some well intentioned copy editor did not change this sentence's meaning unwittingly to make it flow better.