Homotopy theory of function spaces is a healthy subfield of homotopy theory. See e.g. recent Oberwolfach report from a meeting on the subject.

If you share what $X,Y$ you are interested in, I may be able to say more, even though I am by no means an expert.

EDIT: In response to your edit, suppose $\Sigma$ is a compact Riemann surface, $G$ is a compact Lie group, and let's study $k$th homotopy group of $C^0(\Sigma, G)$. Actually, most of the discussion below applies to general $X,Y$ that are, say, compact manifolds or even finite CW-complexes, but let's stick to the case of interest. There is an well-known homeomorphism $C^0(S^k,C^0(\Sigma, G))\cong C^0(S^k\times\Sigma, G)$ given by the adjoint, and since any homeomorphism preserves path-components, we can identify $\pi_k(C^0(\Sigma, G))$ with $[S^k\times \Sigma, G]$, the set of homotopy classes of maps from $S^k\times \Sigma$ to $G$. Smashing $S^k\vee\Sigma$ inside $S^k\times\Sigma$ to a point gives $S^k\wedge\Sigma$, which is the $k$-fold suspension $S^k\Sigma$ of $\Sigma$. The cofiber sequence of the quotient map $S^k\times\Sigma\to S^k\wedge\Sigma$ is an exact sequence, namely, $$[S^k\Sigma, G]\to [S^k\times\Sigma, G]\to [S^k\vee \Sigma, G]=\pi_k(G)+[\Sigma, G].$$ Let's suppose that $G$ is simply-connected, so that $[\Sigma, G]$ is a point (as any Lie group has trivial $\pi_2(G)$ and $\Sigma$ is $2$-dimensional). Then the above cofiber sequence is an exact sequence of groups (not just sets). In trying to compute $[S^k\Sigma, G]$ it helps to recall that $G$ is rationally homotopy equivalent to the product of odd-dimensional spheres, so rationally $[S^k\Sigma, G]$ is the product of $[S^k\Sigma, S^m]$'s where $m$ is odd, and of course $[S^k\Sigma, S^m]=[\Sigma, \Omega^kS^m]$. This should allow you to do some computations.

Finally, I wish to comment that the inclusion $C^\infty(X,Y)\to C^0(X,Y)$ is a weak homotopy equivalence i.e. it induces an isomorphism of homotopy groups as any smooth map of a sphere/disk is homotopic to a nearby smooth map. A result of Milnor says that $C^0(X,Y)$ is homotopy equivalent to a CW-complex, provided $X$ is a finite CW-complex (if memory serves me). I am not sure why the same is true for $C^\infty(X,Y)$ so at the moment I do not know if the above inclusion is a homotopy equivalence.

Homotopy theory of function spaces is a healthy subfield of homotopy theory. See e.g. recent Oberwolfach report from a meeting on the subject.

If you share what $X,Y$ you are interested in, I may be able to say more, even though I am by no means an expert.

EDIT: In response to your edit, suppose $\Sigma$ is a compact Riemann surface, $G$ is a compact Lie group, and let's study $k$th homotopy group of $C^0(\Sigma, G)$. Actually, most of the discussion below applies to general $X,Y$ that are, say, compact manifolds or even finite CW-complexes, but let's stick to the case of interest. There is an well-known homeomorphism $C^0(S^k,C^0(\Sigma, G))\cong C^0(S^k\times\Sigma, G)$ given by the adjoint, and since any homeomorphism preserves path-components, we can identify $\pi_k(C^0(\Sigma, G))$ with $[S^k\times \Sigma, G]$, the set of homotopy classes of maps from $S^k\times \Sigma$ to $G$. Smashing $S^k\vee\Sigma$ inside $S^k\times\Sigma$ to a point gives $S^k\wedge\Sigma$, which is the $k$-fold suspension $S^k\Sigma$ of $\Sigma$. The cofiber sequence of the quotient map $S^k\times\Sigma\to S^k\wedge\Sigma$ is an exact sequence, namely, $$[S^k\Sigma, G]\to [S^k\times\Sigma, G]\to [S^k\vee \Sigma, G]=\pi_k(G)+[\Sigma, G].$$ Let's suppose that $G$ is simply-connected, so that $[\Sigma, G]$ is a point (as any Lie group has trivial $\pi_2(G)$ and $\Sigma$ is $2$-dimensional). Then the above cofiber sequence is an exact sequence of groups (not just sets). In trying to compute $[S^k\Sigma, G]$ it helps to recall that $G$ is rationally homotopy equivalent to the product of odd-dimensional spheres, so rationally $[S^k\Sigma, G]$ is the product of $[S^k\Sigma, S^m]$'s where $m$ is odd, and of course $[S^k\Sigma, S^m]=[\Sigma, \Omega^kS^m]$. This should allow you to do some computations.

Finally, I wish to comment that the inclusion $C^\infty(X,Y)\to C^0(X,Y)$ is a weak homotopy equivalence i.e. it induces an isomorphism of homotopy groups as any map of a sphere/disk is homotopic to a nearby smooth map. A result of Milnor says that $C^0(X,Y)$ is homotopy equivalent to a CW-complex, provided $X$ is a finite CW-complex (if memory serves me). I am not sure why the same is true for $C^\infty(X,Y)$ so at the moment I do not know if the above inclusion is a homotopy equivalence.

Homotopy theory of function spaces is a healthy subfield of homotopy theory. See e.g. recent Oberwolfach report from a meeting on the subject.

If you share what $X,Y$ you are interested in, I may be able to say more, even though I am by no means an expert.

Homotopy theory of function spaces is a healthy subfield of homotopy theory. See e.g. recent Oberwolfach report from a meeting on the subject.

If you share what $X,Y$ you are interested in, I may be able to say more, even though I am by no means an expert.

EDIT: In response to your edit, suppose $\Sigma$ is a compact Riemann surface, $G$ is a compact Lie group, and let's study $k$th homotopy group of $C^0(\Sigma, G)$. Actually, most of the discussion below applies to general $X,Y$ that are, say, compact manifolds or even finite CW-complexes, but let's stick to the case of interest. There is an well-known homeomorphism $C^0(S^k,C^0(\Sigma, G))\cong C^0(S^k\times\Sigma, G)$ given by the adjoint, and since any homeomorphism preserves path-components, we can identify $\pi_k(C^0(\Sigma, G))$ with $[S^k\times \Sigma, G]$, the set of homotopy classes of maps from $S^k\times \Sigma$ to $G$. Smashing $S^k\vee\Sigma$ inside $S^k\times\Sigma$ to a point gives $S^k\wedge\Sigma$, which is the $k$-fold suspension $S^k\Sigma$ of $\Sigma$. The cofiber sequence of the quotient map $S^k\times\Sigma\to S^k\wedge\Sigma$ is an exact sequence, namely, $$[S^k\Sigma, G]\to [S^k\times\Sigma, G]\to [S^k\vee \Sigma, G]=\pi_k(G)+[\Sigma, G].$$ Let's suppose that $G$ is simply-connected, so that $[\Sigma, G]$ is a point (as any Lie group has trivial $\pi_2(G)$ and $\Sigma$ is $2$-dimensional). Then the above cofiber sequence is an exact sequence of groups (not just sets). In trying to compute $[S^k\Sigma, G]$ it helps to recall that $G$ is rationally homotopy equivalent to the product of odd-dimensional spheres, so rationally $[S^k\Sigma, G]$ is the product of $[S^k\Sigma, S^m]$'s where $m$ is odd, and of course $[S^k\Sigma, S^m]=[\Sigma, \Omega^kS^m]$. This should allow you to do some computations.

Finally, I wish to comment that the inclusion $C^\infty(X,Y)\to C^0(X,Y)$ is a weak homotopy equivalence i.e. it induces an isomorphism of homotopy groups as any smooth map of a sphere/disk is homotopic to a nearby smooth map. A result of Milnor says that $C^0(X,Y)$ is homotopy equivalent to a CW-complex, provided $X$ is a finite CW-complex (if memory serves me). I am not sure why the same is true for $C^\infty(X,Y)$ so at the moment I do not know if the above inclusion is a homotopy equivalence.