As topological spaces, all of $\text{Spec}(k), \text{Spec}(k(x)), \text{Spec}(k[x]/(x^2))$ and $\text{Spec}(k(x_1,\cdots,x_n))$ are all homeomorphic, since they are all one point-spaces.
However, as schemes they are different. We have been told to think of $\text{Spec}(k[x]/(x^2))$ as a point together with a tangent direction, but how do we think of, for example, $\text{Spec}(k(x_1,\cdots,x_n))$?
My guess would be that a good way to think about $\text{Spec}(k(x_1,\cdots,x_n))$ is as a point together with a point in $\mathbb{A}_k^n$.
(inspired by the fact that $\text{hom}(Spec(k),Spec(k(x_1\cdots,x_n))=\mathbb{A}_k^n$ (at least for algebraically closed k))