Going from a stress-energy distribution to the motion of test particles is a long and arduous process in general.
To see how it works, you need to understand that the left-hand side of the Einstein equation is just a very concise shorthand for a function of the metric. You can back out this dependence using the standard definitions
\begin{align}
G_{\mu\nu} & := R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}, \\
R & := g^{\mu\nu} R_{\mu\nu}, \\
R_{\mu\nu} & := R^\lambda{}_{\mu\lambda\nu}, \\
R^\rho{}_{\sigma\mu\nu} & := \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma}, \\
\Gamma^\sigma_{\mu\nu} & := \frac{1}{2} g^{\sigma\rho} \left(\partial_\mu g_{\nu\rho} + \partial_\nu g_{\rho\mu} - \partial_\rho g_{\mu\nu}\right).
\end{align}
At each step you can pause and ask for an interpretation. For instance, $\Gamma^\sigma_{\mu\nu}$ is the set of components that describe the difference between covariant differentiation and partial differentiation in your coordinate system, and $R$ is related to a local "radius of curvature" for your manifold.
As you can see from searching for partial derivatives, $G_{\mu\nu}$ will be a (probably horrendously nonlinear) function of the metric and its first and second derivatives. The Einstein equation, then, is just a partial differential equation that gives you the metric if all you know is the stress-energy.
After you have the metric, you can plug in to the geodesic equation
$$ \frac{\mathrm{d}^2x^\mu}{\mathrm{d}\lambda^2} + \Gamma^\mu_{\rho\sigma} \frac{\mathrm{d}x^\rho}{\mathrm{d}\lambda} \frac{\mathrm{d}x^\sigma}{\mathrm{d}\lambda} = 0. $$
Okay, so maybe you don't need to get all the way down to the metric - knowing the connection coefficients is sufficient. But we can always make things more complicated again by adding non-gravitational accelerations to the right-hand side of the geodesic equation. In any event, this is another differential equation you would solve to get, together with the initial conditions, the trajectory of a particle.
In practice, you would have a computer do all of this numerically (which is still much harder than it sounds), or you would be handed values of $T_{\mu\nu}$ that have enough symmetry (perhaps the tensor is diagonal, or constant in spacetime, or spherically symmetric) to make the differential equations tractable.
