I guess it depends on your specific interest also. I certainly agree with Long's suggestion on Bruns and Herzog above.
From a more geometric perspective... A good source for local cohomology / duality stuff is Hartshorne's book "Local cohomology" based on Grothendieck's seminar (I think)
It's a lot more geometric than Bruns and Herzog.
You can also move from there to "Residues and Duality" if you'd like (and there are other sources for that as well, Brian Conrad's book, Lipman's notes, etc.). Coming from a more geometric perspective originally myself, I didn't really get Bruns and Herzog chapter 3 until I did this.
For Eisenbud's book, perhaps you should take it chapter by chapter. Many chapters don't really rely on anything and can be read out of context. This makes it a very valuable reference.
