I want to learn the book, but it seems that I should have some background on harmonic analysis, Lie Groups and measure theory. Can you give some references?
To discuss generally first, the book was written up by C. P. Ramanujam, and he was more conscientious than usual in trying to tie down Mumford's lectures to existing references. Still, it is quite hard to sort out the exact prerequisites.
The analytic theory and theta-functions required in the first chapter stands rather aside from the rest, which is mostly in the EGA/SGA vein. You do need some Fourier theory for Chapter I; the style given is a bit Bourbaki-oriented, I recall, but there are alternatives (for example, take certain things for granted, and/or read another text on the analytic theory).
As for the rest, I remember finding the homological algebra prerequisites steep. Hartshorne's text will take you some way; but the spectral sequences and mapping cylinders rather expect expertise. Since the "Great American Sheaf Theory Book" isn't yet written, I'm not quite sure what to recommend. Probably a more modern treatment of the duality theory post-Mukai would be good to look at anyway, rather than obsessing over each of Mumford's proofs.