I'd like to begin experimenting with some IC's and other basic electronic parts. But I'm new at it. My first step is to collect some of the parts I think I will be needing.

The parts (IC's) I've collected so far are as follows:

6502 CPU
6522 VIA
Z80 CPU
Z80 PIO
3to8 Decoder IC
64k x 4 Ram (handfull of these)

But now I need some sort of ROM. In particular some EEprom. What I'm currently looking at are 2864 8k x 8 EEprom's. But I have a question on it. Is there any major difference between the 2864 EEprom and something like the 28HC64 EEprom. Is there any reason either would not be compatible with the parts I've already purchased? Is one more sensitive then the other? In other words, is the 28HC64 much harder to handle because of the possiblity of short circuiting the ROM with static electricty.

Generally all the LSI is static-sensitive, even those parts that don't have a "C" in the identifier. Your '138, if it's 74LS138, is not particularly static-sensitive, but everything else you mention is. These parts do have protection diodes built in, but they can only handle so much. If you're going to work with this stuff at all, you should:
A. have an antistatic mat like Jameco sells and connect it to the grounded chasis of a piece of equipment on your workbench,
B. have the tip of the soldering iron connected to the mat or something that dissipates static, and
C. touch something grounded to discharge the static in your body before you touch static-sensitive parts.

I've never used more than this (like the writst strap, antistatic lab coat, antistatic flooring, humidifier, air ionizer, finger cots, or anything like that), and I've never damaged these parts with static. I've seen plenty of other people damage them by being careless though.

Any compatibility issues between 28HC64 and 2864(without the HC). I'm a little confused on the difference between the two.