We were just there this summer, it was my wife's favorite park.
If you do the Narrows, which is just incredible, stop by Zion Outfitters, just before the park entrance, and at least get a walking stick. They help a LOT w/ balance and hiking poles aren't quite good enough because they can get bent between the rocks. We also got the neoprene socks/shoes since it was our first time and the water was a little chilly in June. You can maybe skip this stuff if you've done it before, but we wanted to be safe our 1st time and with kids. It's about $30 for the bundle.
Angels Landing does require a permit, but you can hike about half (or more) of the way up before permits are enforced. The view is still incredible from that spot. You still hike past Walter's Wiggles and see the valley before the permit area.
Both of these are good options. There is no other hike like the Narrows though.
There is another area outside the park called Kanarra Falls, which also requires permits but they are cheap and just meant for crowd control. Google it, we would have gone there if we had more time. Some people like it more than the park because it's less crowded and it's basically a slot canyon w/ an awesome waterfall thru it.