Firstly, we solve the Einstein field equations to obtain the metric tensor. After that, we solve the geodesic equations to obtain the geodesics. Is it like this? What is the brief description of solving the Einstein field equations?
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It depends on what your starting point is. If you are given the matter distribution (which is a reasonable starting point), then you can deduce from this the stress tensor. Using the stress tensor (which is the RHS of the EFE), we can deduce the metric. Finally, using the metric, we can deduce the geodesics.
