Whenever you can win the lottery
Lost Creek and Mazama Campgrounds
The Wave is one of the coolest single features I've seen through my journeys in America. It's undulating sandstone rings are somewhat mesmerizing. If it weren't for a wind whipping sand that reminded me of my nights spent at Lake Powell I may have lost track of time and forgotten to hike back to my bicycle before the sun set. There are two things that are tricky to the wave:
1) Getting a permit (either you need to plan well in advance and secure permits online or you have to win a permit through a lottery system)
and
2) Finding the Wave and finding your way back
Other than that, it's great. The lottery is only tricky in that it's getting a permit via a game of chance - it's actually really fun participating in it with all the other hikers that hope to hike to the Wave the following day. I'd probably reccomend a GPS for the hike. The map that the Rangers give to you is very helpful in getting to the wave, but you're easily disoriented on the way back and the sandstone buttes, and sagebrush hills tend to look the same each one from the next.
If you don't camp out and use all of your daylight hours at the wave it's highly reccomended to make the short trip through Wire Pass and into Buckskin Gulch. Wire Pass is one of Utah's many slot canyons and it can be equally as astonishing as the Wave. I loved this place and I will be planning to return to hike the Gulch from beginning to end as it's one of the top 10 hikes in the world (per who, I don't know, but the Rangers said so).