It is true that the primary object of interest is $H^0$—global sections. I will give you four reasons for the importance of higher cohomology: use of exact sequences, the Riemann-Roch problem, the Cousin problem, and the GAGA theorems. I conclude the answer with discussing the problem of representing geometrically cohomology classes.

1. Exact sequences

As you observed, cohomology reveals useful when some exact sequences of sheaves do not lead to exact sequences at the level of global sections – something we now understand as non-vanishing of a first cohomology group. Some classical restriction theorems of Algebraic geometry can be seen as stating the vanishing of a first cohomology group. But once you accept exact sequences with 6 terms involving $H^0$ and $H^1$, why don't go further and extend it with $H^2$, etc.?

2. The Riemann-Roch problem

Historical 19th century problems of algebraic geometry asked to construct holomorphic/meromorphic functions with prescribed zeroes/poles on the complex plane $\mathbf C$, on Riemann surfaces, or on $\mathbf C^n$, or on complex algebraic manifolds. That's the origin of the Riemann problem and the Riemann inequality : if $D$ is a divisor on a Riemann surface $X$ of genus $g$, meromorphic functions $f$ with poles at most $D$ are global sections of a sheaf $\mathscr O_X(D)$, and the Riemann inequality states that $$ \dim (\Gamma(\mathscr O_X(D)))\geq \deg(D)+1-g.$$ The Riemann inequality has been made more precise thanks to Roch, leading to the Riemann-Roch equality: if $K$ is a canonical divisor on $X$, then $$ \dim (H^0(\mathscr O_X(D))) = \deg(D)+1-g + \dim (H^0(\mathscr O_X(K-D)).$$

The picture gets messier on surfaces where Italian geometers had proved an inequality of the form $$ \dim(H^0(\mathscr O_X(D)))+\dim(H^0(\mathscr O_X(K-D))) \geq \frac12 D\cdot (K-D) + 1 + p_a, $$ wher $p_a$ is the arithmetic genus. As Hartshorne writes (Algebraic geometry, p. 363) : The difference is called the superabundance, because, before the invention of cohomology, it was the defect of validity of the corresponding equality. Having a natural equality is the content of the Riemann-Roch theorem for surfaces offers a more precise theorem involving Euler-Poincaré characteristics $$ \chi(\mathscr O_X(D)) = \dim(H^0(\mathscr O_X(D)))- \dim(H^1(\mathscr O_X(D))+\dim(H^2(\mathscr O_X(D))) = \frac12 D\cdot (K-D) + \chi(\mathscr O_X), $$ which implies the former formula thanks to the duality theorem $$ H^2(\mathscr O_X(D)) \simeq H^0(\mathscr O_X(K-D))^\vee. $$ A similar interpretation holds in the case of Riemann surfaces, $$ \chi(\mathscr O_X(D))=\dim(H^0(\mathscr O_X(D)))-\dim(H^1(\mathscr O_X(D)) = \deg(D)+1-g, $$ and the duality theorem $$ H^1(\mathscr O_X(D)) \simeq H^0(\mathscr O_X(K-D))^\vee. $$

The generalization in higher dimension has been proved by Hirzebruch and Grothendieck, but is still of the same form: a formula for the Euler-Poincaré characteristic $\chi(\mathscr O_X(D))$ in terms of intersection theory and characteristic classes.

It has to be complemented with duality theorems (Serre, Grothendieck) and vanishing theorems (Serre, Kodaira), which allow to express the Euler-Poincaré characteristic in terms of more concrete objects, such as $H^0$.

3. Cousin problems

The Cousin problem asks to construct meromorphic functions with prescribed poles on the complex plane. If one tries to generalize it, one observes that its possibility lies in the vanishing of some cohomology groups, such as $H^1(X,\mathscr O_X^\times)$.

4. The GAGA theorems

Here GAGA is not a pop star, but an acronym for Géométrie algébrique et géométrie analytique, the title of a famous paper by Serre. Some comparison theorems between those two geometric theories were well known: for example, any meromorphic function on the projective space $\mathbf P^n(\mathbf C)$ is a rational function, and any analytic closed subvariety of $\mathbf P^n(\C)$ can be defined by polynomials (Chow's theorem). Serre's generalization is sheaf theoretical: it says that the functor of analytification of algebraic coherent sheaves leads to an equivalence of categories between the categories of algebraic and analytic coherent sheaves. Its proof is cohomological and establishes that the analytification functor preserves cohomology groups, a fact that Serre proves by decreasing induction, starting with cohomology of high degree, where the result is trivial because both cohomology groups vanish in degrees strictly larger than the dimension. The most important statement, that of global sections — $H^0$ —, is obtained at the last step of the induction, hence is in some sense the most difficult.

5. The quest for a geometric interpretation of cohomology

Maybe, a difficulty of cohomology lies in the lack of a geometrical interpretation.

For $H^1$, this is quite easy to achieve: $H^1(X,\mathscr F)$ parameterizes principal homogeneous spaces under $\mathscr F$; as we saw above, this interpretation is already useful in the study of the Cousin problem.

The description is much more complicated for $H^2$ — it is the topic of nonabelian cohomology, initiated by Grothendieck and Giraud.

I do not know however of similar geometric interpretations in higher degrees.