If you are staying in Moab and want to see some arches not in the park, you can go see one of the largest in the world at Negro Bill Canyon. Located just a few miles up highway 128, this is a popular hike so the earlier you get there, the better chance you have at finding parking.
Note: The sign for Negro Bill Canyon has the word negro painted out. Apparently the name is offensive to some, so if you are looking for something that says Negro Bill Canyon it will only say Bill Canyon.
The hike is relatively flat and is 2.5 miles one way (5 total).
When we did this hike it was our last day in Moab. We arrived early and found parking, luckily, but by time we left the parking lot it was almost full. Since we are crazy/ parents of the year (according to some) we brought along our new baby.
The hike is relatively easy and is a great kid hike. Dogs are also allowed. The trail zig zags up a large box canyon with a pleasant stream that runs down the center of it. Total elevation change is about 300 feet. The one steep part of the trail is near the very end. Just as you cross the river for the last time, the trail makes a steep climb to the top of a large hill. From here the trail is flat to the arch.
The crowning jewel of the hike is Morning Glory Bridge. This Arch is 243 feet as in number 7 of worlds largest arches.
From the base of the arch is a natural spring that you can refill your water from.
Something cool about this hike. It can be taken from the top down. There is an outfitter there in Moab that will take you and you can rappel down between the arch and finish the hike.