If I specify a metric tensor in terms of a choice of bases in the co-tangent space at a given point in a manifold, how much information does the metric tell me about the curvature tensor at that point? Can I calculate the curvature tensor entirely from the metric tensor?
Yes you can, but you need to know the metric up to the second derivative, so the metric in a neighb. of the manifold. You can do that for the Levi-Civita connection, since it is completely defined by the metric $g$.