Let $x_1, \ldots, x_n$ be your points on the sphere. Choose an $n+1$-th point $z$ on the target sphere, and paths from $z$ to each one of the $x_i$s. By moving along these paths you can construct a homotopy from the space of maps that fix the $x_i$s to the space of maps that send all the points $x_i$ to $z$.
This latter space is the same as the space of pointed maps from the quotient space of $S^2$ by $n$ points to $S^2$. Easy exercise: this quotient space is homotopy equivalent to the wedge sum $$S^2\vee \bigvee_{n-1}S^1$$ It follows that the space of maps that you are asking about is homotopy equivalent to $$\Omega^2S^2\times (\Omega S^2)^{n-1}$$ Here $\Omega^k X$ denotes the space of pointed maps from $S^k$ to $X$.
The set of path components of this space is the same as $\pi_2(S^2)$, which is $\mathbb Z$. Its higher homotopy groups can be written in terms of those of $S^2$.