The helmet is a necessary piece of safety gear for going underground. Virtually anyone who caves for long enough will make the mistake of standing up in a chamber where the roof isn't quite high enough, and giving themselves a bad bump on the head. Falling debris is also a serious issue. Any helmet should have interior suspension, to provide some absorption for a blow. The helmet should also have a chin strap. Getting a proper caving helmet, with mounting points for a headlamp, is a good idea. However, a construction worker's helmet is good enough in the short term.
Another necessity for caving is a minimum of two light sources. One of these is always your headlamp. The other is having a back-up hand-held flashlight. You should pack at least one full set of replacement batteries for both the flashlight and the headlamp, and it is not a bad idea to try to make those batteries interchangeable if you can. A replacement light bulb for the headlamp is also a good idea, since that is your primary light source and much more likely to take some bad hits than your flashlight. A final lighting back-up would be to bring along some glow sticks as a light source that can be activated easily, even while fumbling around in the dark.
Caves have a predictable temperature of 54 degrees Fahrenheit, and the environment down there is often damp. You may also encounter areas of liquid mud, as well as underground pools, streams or water dripping out of the cave ceiling. Your clothes need to be warm, and the outer layer needs to be water-resistant. Also, caving involves a lot of crawling around in the mud and stone, so the clothes should be tough. Many cavers choose a workman's jumpsuit or overalls plus a light jacket as an outer layer, and avoid cotton fabrics for the layer of clothing underneath. That especially applies to socks. Cotton offers no insulation at all when it gets wet, and wearing your cotton gym socks pretty much guarantees cold feet in a cave. A good pair of hard knee pads is also part of the essential caver's apparel, as are a pair of water-resistant or waterproof hiking boots.
Items and Supplies
Other strongly recommended caving items include a plastic bag to serve as a waterproof liner for your backpack or other packs, a compass, a cave map and a whistle to blow if you should get lost. Someone in the party should also bring a coil of good climbing rope, even if you don't need to rappel into the cave. A standard safety precaution for the lead person checking out a belly-crawl tunnel is to tie a rope around their waist or ankle, so they can be pulled out if the tunnel proves to be a dead end and turning around is impossible. Also remember that despite the low temperature, caving is hard work. Always bring along a canteen or bottle of water and a good, high energy snack.