I am considering reading one of 'Algebraic curves and Riemann Surfaces' by Rick Miranda or 'Lectures on Riemann Surfaces' by Otto Forster. Which one of these is more advanced and comprehensive ? What are the differences in the approaches of these two books ?
I know Forster's book quite well, having taught out of a good portion of it a few times. It is extremely well-written, but definitely more analytic in flavor. In particular, it includes pretty much all the analysis to prove finite-dimensionality of sheaf cohomology on a compact Riemann surface. It also deals quite a bit with non-compact Riemann surfaces, but does include standard material on Abel's Theorem, the Abel-Jacobi map, etc. I would also recommend Griffiths's Introduction to Algebraic Curves — a beautiful text based on lectures.
I think this post is almost a duplicate. See the following:
I do recommend the recent published book by Donaldson on this subject. It is really interesting to read.