I think that the quickest route to the effective action actually isn't through string amplitudes, but through the beta functions. The conditions for worldsheet conformal invariance are equivalent to the spacetime equations of motion, and from these you can infer an on-shell effective action. This is all you can hope for in string theory (or any theory of quantum gravity), which isn't well-defined off-shell.
As for references, Polchinski does the bosonic case. These computations are easiest in dim reg, but his scheme is fine also. I think the original analysis was by Callan et. al, http://adsabs.harvard.edu/abs/1985NuPhB.262..593C.
There are also really fun technicalities that come up in higher genus corrections. These were pointed out by Fischler and Susskind http://inspirehep.net/record/17879, and expanded on in a series of papers by Fradkin and Tseytlin, and Callan et. al.