The number one thing to never leave home without. Cavers like to have three lights per person. In urban exploration one or two per person usually works. Just be sure the group has more than one light. High-candlepower spotlights are often useful. Many adventurers also use headlamps; being as hands-free as possible is a good guideline in all equipment matters.

Pretty self-explanatory. Minimally, carry one spare set. Also, a spare for your camera/flash isn't a bad idea.

Preferably some type of sturdy work glove. It'll only take one time of grabbing a fiberglass-coated pipe or putting your hand down in a puddle of caustic goo to teach you this on your own; save the learning experiences for suckers and wear gloves.

From cutting ropes to popping latches (because you forgot your keys) to digging out that nasty metal splinter you got from leaning against an old mill press (try wearing some gloves next time), a knife is indispensable.

It can be a backpack or something smaller like a hip-pack, depending on your needs; the key is hands-free gear storage. If nothing else you want a way to carry your group's spare lights and batteries.

Consider shorts for wet work, long pants for anything else -- even in warm weather. Having as much skin as possible protected from scrapes, burns, toxic gunk, etc. is a good idea. In this vein, it's common to wear an overshirt even in hot steam tunnels. Most people stick with jeans, but for draining keep in mind that synthetics (fabrics starting with "poly-") dry much more quickly than cotton/denim. As for shoes, use your common sense. If you're getting wet, wear shoes you don't mind soaking. If you're in an abandoned building, wear something thick-soled in defense of stray nails and rusty machinery. Consider traction. Some people choose sandals for drain work; some don't feel comfortable sinking their feet directly in rank funk. Personal preference.