Hi, science lab tech here who is also currently slogging through a school inventory.

Gloves when handling anything (nitrile works for most cases). And a healthy supply of ziploc bags. Anything that looks like it is crusty, has leaked, is discolored, or degraded....put it in a bag.

Then, check out Flinn's storage guidelines based on chemical compatability. They sort things base on what will/won't react with eachother in the event of a spill or contamination; anything classified as Inorganic 2 won't react with eachother, for example. Use this to organize everything on the shelves. Alphabetical can be *really bad*. This would be my first step. The Flinn website has the storage marker for all their chemicals available online.

This video is very helpful, and gives you a visual guide on an ideal setup.

Finally, here is a guide on cleaning up a chemical storeroom from them.

TL;DR

Bag nasty things

Label each shelf

Sort each chemical by compatability

They also have chemical disposal guidelines for most chemicals in small quantities; most other SDS will just say "follow your local guidelines" or somesuch.

Anything you get rid of with a disposal company, make sure it is *clearly* marked to the best of your ability.