You can perform an INI manually by pressing the button on the faceplate of the call processor card.
An INI is a warm start (think of pressing CTRL-ALT-DEL) on an older PC to reboot it.
A sysload is a cold start (think of pulling the power from a PC, plugging it back in and turning it back on).
Generally speaking, a well-maintained PBX in good working order will run for years without ever needing to be INI'ed or a sysload unless the release notes for a software patch require it. If you're finding yourself having to reboot the PBX on a semi-regular basis, it has problems.
INI's and sysloads that happen out of the blue are a cry for help. If you're seeing those happening - call your vendor... don't bother trying to diagnose the issue. It's not a customer DIY project unless you have years of experience under your belt with these things... and if you need software patches and the such, you'll need to go through a vendor anyway as Avaya doesn't issue software to customers.
But the fact you're asking about INI's indicates you're having problems that a vendor should be looking at first. If your PBX is covered under a maintenance contract with a vendor, you're already paying for the service.
There's not really a shut-down procedure for a PBX. Simply disconnect power from the PBX. Tip: don't throw the switches on the power supplies in the cabinets... if you're running an Option 11C - just pull the power cords from the wall, or if you're running a larger system (61C, 81C, etc...) throw the breaker for the core shelf in the pedestal (left to right - breakers are for the pedistal fan, then the 1st, 2nd, 3rd and 4th shelves from bottom to top). There usually isn't a need to shut down all the shelves unless you're running into some problems where you need to isolate parts of the system as a troubleshooting step - in most cases just the core shelf itself. Power supplies tend not to like being shut down with the convenient power on/off switches on their faceplate - they may not come back up when you go from Off to On. Just experience from the field speaking.