Go everywhere twice. This is particularly true for places that receive sunlight through small openings. A few hours pass and the whole place looks completely different. The first time I came to surprise arch it was noon and harsh overhead light washed out the sky and make no interesting shadows on the walls. When I returned, the sun was low in the west and illuminated the far wall just enough to still allow the low-lights to remain visible in the foreground.

It also may have helped that when I returned I came alone. My initial experience was marred by the presence of a group of "jeep enthusiasts" who had been tear-assing in and around Moab all day in a big convoy of 6 jeep wranglers. They really loved their jeeps and could not go anywhere without them, even when they were all going to the same place each individual had to pilot his own hulking gas-guzzler. They even seemed to always travel in the same order. The red one with the biggest tires must have been the leader, kind of like the ibex with the biggest horns. Way in the back of the convoy was the teal stock wrangler, clearly the omega-male. This herd of offroaders seemed to be following me everywhere for a couple of days.