You need a PCI card with one or two IDE ports on, in order to support 5 IDE devices.
The conventional way to connect IDE devices is to put your two Hard Drives on the Primary IDE, and your other drives on the secondary or plug-in card.
If you get a card with two more IDE channels, though, I would suggest the following
IDE Primary - Hard Drive (System drive)
IDE Secondary - CD-RW as master - ZiP drive as Slave
PCI IDE1 Master - Second hard drive
PCI IDE2 master - New DVD-RW Drive
"My gigabyte board supports A or B drives but does that include a ZIP drive?" .... I don't understand. A and B drives are a floppy interface - A ZiP drive is IDE, different altogether.