Water
There is no hiking in Utah's canyon country without water. You must carry large amounts or know that there are water sources or seeps you can count on. As in most places, found water should be purified. A day in the red rock heat desiccates your body, often without symptoms, and drinking 4 liters a day is not an inflated goal.
Sun Protection
Sunscreen that is rated 30 SPF provides almost 97 percent protection from that pesky ultraviolet radiation; higher SPFs aren't substantially more effective. The trick is to cover all exposed surfaces and to reapply often. Some frequently forgotten spots are (1) feet (if you're wearing sandals), (2) under chin, (3) ears and (4) hands. Your lips should be smeared with at least 30 SPF lip balm. Polarized sunglasses protect eyes and reduce glare, and an amply brimmed hat keeps sun off your face and reduces body heating. Always carry a long-sleeved shirt and long pants that block the sun's access to your body.
Boots
Footwear in canyon country depends on your particular hike. Sneakers or water sandals are great for walking down washes or streams through canyons and may be used on slickrock surfaces, but when the going gets rocky, brushy or steep, a good pair of ankle-high boots with friction-pattern soles is necessary.
First Aid Kit
On these trails, you may very well need to treat cuts and bruises or even broken limbs, so a high-end kit is advisable. It's also a good idea to be conversant in treatment for heat exhaustion and to carry a snake-bite kit.
Maps and Knowledge
This country is complicated. That's why names like the "Labyrinth" and the "Maze" pop up when you Google "Utah." The canyon systems branch and loop incessantly, so a well-understood map might save you. Canyon waters can rise rapidly during flash floods, so it's judicious to know your route and its escape valves.
Climbing Equipment
If you have done your homework, you'll know whether you need to be packing ropes, harnesses, carabiners or other climbing gear. There is a good bit of technical climbing in the canyons, especially entering and exiting.
Communication
Should you need to be rescued, you will want every possible means of reaching other people. Cell phones are great, but service may disappear in canyons. Walkie-talkies between team members who are not within voice distance are useful, and a whistle can be heard for long distances.