For the Narrows hike in Zion you'll want hiking boots that support your ankle with a good grip at the bottom, neoprene socks (they protect your feet from getting wet), a walking stick, and possibly waterproof pants or a bib, depending on what season it is. If you don't have that gear and will only use it this one, it's more cost-effective to rent gear than purchase it. Zion Outfitter in Springdale rents Narrows gear for every season and every size.  Experienced trekkers will tell you that rentals are a waste of money, and you can buy your own gear for almost the same price. That's up to you, but you have options either way. Whatever you do, don't enter the Narrows with opened-toed shoes or tennis shoes.