Ideally, one should have three good independent sources of light, any one of which could be used as a primary light source. The hard-hat not only protects your head, but it also carries your lamp, leaving both hands free at all times. Get one that fits properly, as a loose hard-hat is more of a hindrance than a help. The hard-hat should also have a secure chinstrap. Whitewater boating helmets are suitable protection, but do not have a lamp mounting bracket.
Sneakers are adequate, but any hard-sole shoe or boot is sure to cause an injury.
Note that your shoes will get very muddy. Given the choice between wearing expensive hiking boots or sneakers, I would wear the sneakers on my first trip. Wear wool socks for warmth. Rugged long-sleeve clothing should be worn. Coveralls are nice.