Is there some software or technique that would allow me to visualize a (2 dimensional) Riemannian manifold given its metric tensor and second fundamental form?
For instance, given $$ g =d\theta^2 +\sin(\theta)^2 d\varphi^2$$ and $$ K_{ab} \sim \begin{pmatrix} a^2 & 0 \\ 0 & \sin(\theta)^2 \end{pmatrix}$$
it would plot some oblong $\mathbb{S}^2$ type object.
Yes There is. Please visit Reconstructing a Surface from its Fundamental Form Coefficients. Just note that the result comes by numerically solving differential equations and the file comes in Maple software format.