Curvature can be very easily pictured using ‘geodesic quadrilateral gaps’, which can be more generally used to recover the torsion tensor, and if the torsion is identically zero, then the curvature tensor, for a manifold equipped with an affine connection.
In the special case of an oriented Riemannian surface $(M,g)$ with its Riemannian connection $\nabla$, this works as follows to pictorially give us the Gaussian curvature $\kappa(P_0)$ at any point $P_0 \in M$. Travel along a geodesic from $P_0$ in the starting direction given by a unit vector $u\in T_{P_0}M$, and take the point $P_1$ on it at a small distance $s$ from $P_0$. Turn left in $90$ degrees, and follow the geodesic in that direction for the same distance $s$ to arrive at a point $P_2$. Iterate the left turn and the travel along the geodesic for distance $s$ twice more, to successively arrive at points $P_3$ and $P_4$. If the surface was flat, and $s$ small enough, then we would have traveled along a closed geodesic quadrilateral and arrived back at the starting point, that is, $P_4 = P_0$. But if the curvature is non zero, then the vector $P_4 - P_0$ (which you can define in terms of a local smooth embedding of $M$ in a higher dimensional vector space) is non-zero, and satisfies the following formula. Let $v\in T_{P_0}M$ be the vector such that $(u,v)$ is a right-handed orthonormal basis for $T_{P_0}M$. Then $$\lim_{s\to 0}\, {P_4 - P_0 \over s^3} = {\kappa(P_0) \over 2}(u - v).$$
More generally, let there be give a pair $(M,\nabla)$ where $M$ is a smooth manifold and $\nabla$ is a connection on $TM$. Consider any $P\in M$ and a pair of vectors $u,v \in T_PM$. From the triple $(P,u,v)$ and a small real number $s$, we can make a new triple $(P',u',v')$ as follows. Take the geodesic from $P$ with starting tangent vector $u$, and let $P'$ be the point on it where the affine parameter takes the value $s$ (where the parameter has value $0$ at $P$). Let $u',v' \in T_{P'}M$ where $u'$ is parallel transport of $v$ and $v'$ is $(-1)$-times the parallel transport of $u$ along this geodesic. Starting with a triple $(P,u,v)$ for which $P = P_0$, and iterating the above, we get an open geodesic quadrilateral with vertices $P_0$, $P_1 = (P_0)'$, $P_2 = (P_1)'$, $P_3 = (P_2)'$ and $P_4= (P_3)'$. The quadrilateral is closed if $P_4 = P_0$. But in general, we have the formula
$$\lim_{s\to 0}\,{P_4 - P_0\over s^2} = - T(u,v)$$
where $T(u,v) = \nabla_uv - \nabla_vu - [u,v]$ is the torsion tensor. If the torsion tensor $T$ is identically zero on $M$, then the gap $P_4 - P_0$ is given in terms of the Riemann curvature tensor by the formula
$$\lim_{s\to 0}\,{P_4 - P_0\over s^3} = {1\over 2}R(u,v)(u+v)$$
where by definition $R(u,v)(w) = \nabla_u\nabla_vw - \nabla_v\nabla_uw - \nabla_{[u,v]}w$. The above formula can be ‘inverted’ to recover the curvature tensor when the torsion is identically zero, as the tensor $R(u,v)(w)$ can be recovered uniquely from the tensor $R(u,v)(u+v)$ using the symmetries of $R(u,v)(w)$.
The above results are proved in arXiv:1910.06615, which is written in an expository style.
