In calculus you're sometimes told that the second derivative $y''$ at a critical point determines the curvature of a curve. Studying further we find this is true because the formula for curvature $$k = \frac{y''}{(1 + (y')^2)^{3/2}},$$ when $y' = 0$, is just $y''$. In differential geometry, you can derive the Riemann curvature tensor \begin{align} &R^a_{bcd} = - (\Gamma^a_{bcd} - \Gamma^a_{bdc} + \Gamma_{bc}^e \Gamma_{de}^a - \Gamma^e_{bd} \Gamma^a_{ce}) \\ &[\text{Note the nice} \ -(bcd - bdc + bc^e d_e - bd^e c_e) \ \text{pattern!}] \end{align} by deriving the equation of geodesic deviation (Jacobi fields) on $v^a = \frac{\partial x^a(t,s)}{\partial s}$ on geodesics $x^a(t)$ parametrized by $s$, $x_s(t) = x(t,s)$, $u^a = x^a_t$: $$\frac{D^2 v^a}{Dt^2} = - R^a_{bcd} u^b u^c v^d,$$ which shows the second covariant derivative along a curve determines the curvature along geodesics, just as the second derivative of $y = f(x)$ determines the curvature at a critical point.
Is there an analogue of $k = \frac{y''}{(1 + (y')^2)^{3/2}}$ for the Riemann curvature, that reduces to $R^a_{bcd}$ along geodesics?
Sectional curvature kind of looks like it but this formula reduces to it for orthogonal vectors, no mention of geodesics.
The two notions of curvature that you'd like to compare are conceptually different.
The Riemann tensor describes the intrinsic curvature of a manifold, measuring (roughly speaking) the failure of ordinary flat Euclidean geometry: how geodesics that start parallel diverge or converge, how for a small circle the ratio of circumference to radius deviates from $2\pi$, and so forth. "Intrinsic" means that these notions all make sense within the manifold itself.
The curvature of the graph of a function that you quote as $$ k = \frac{y''}{(1 + (y')^2)^{3/2}} $$ is an extrinsic notion of curvature, which tells you how one manifold (your curve $y=y(x)$ in this case, the "submanifold") is embedded in another (the flat two-dimensional plane here). This notion of curvature tells you, for example, how quickly a geodesic in the ambient space diverges from a geodesic within the submanifold. It's extrinsic because it makes sense only if you refer to the space outside the curve, and how the curve lies in that higher-dimensional space.
To mark the difference, intrinsically a plane or curve or cylinder are all identical (you can make them from a flat sheet of paper), but they differ in extrinsic curvature, which refers to how they're embedded in the ambient three-dimensional space.
The object in differential geometry that generalises $k$ is a tensor sometimes called the second fundamental form. It tells you how, if you start with a vector tangent to a submanifold and parallel transport it by another tangent vector, it moves off the submanifold. Geodesics have vanishing second fundamental form, because they parallel transport their own tangent vector: they're the "straight lines" on the manifold, with no extrinsic curvature.