What is the necessary background to read this book and comprehend it? I've taken Calc I-II, Chem 1A and Bio 1A. Is this enough?
I think you should reverse your approach. Rather than trying to find every prerequisite, start reading sections of PNS and as you encounter something you don't understand, look up the background information. It's an exhaustive work (and well worth reading!) but you're going to find different sections require different levels of background. For example, the section on ion channels requires a bit of E&M physics, and the section(s) on neurotransmitters will make more sense with some basic organic chem.
If you get through some of it and feel like you need more of the basics, a book like Bear's is a bit more accessible.
There's another version of Kandel (it's probably older than you are, but your library might be able to get it) that takes a slower approach.
Basic concepts of biochemistry and some knowledge of general biology (mainly, general cytology) are sufficient (Chem1A and Bio1A would be fine); some specific chapter requires a little more advanced knowledge of molecular biology and human physiology, but nothing "over the top".