El Morro National Monument is located near Zuni and is highly recommended.
The hike through this park takes you on a path around the mesa, on which you can see carvings from past explorers to the area. Past this you hike up onto the mesa, where you get fantastic views of the countryside. On the way back down, you pass a 750-year-old settlement that in many ways remains a mystery. The park can be seen in a matter of hours, and because it's quite far off the beaten path, you don't have to worry about the crowds.
It is VERY important that you be sensitive to the Native culture here. If you plan to go anywhere on the reservation, be sure to check first as certain sites are holy and off-limits to non-Zunis. The people here don't see many outsiders and are naturally curious, so be prepared to attract some attention. They will usually welcome you warmly as a visitor, but they are often quite suspicious of outsiders, particularly the older members of the community. This has been changing some, particularly among the young. Still, be sensitive to the fact that some people here may not trust you and would rather you not spend a whole lot of time there. Again, most of the locals are very friendly, but don't be surprised if not everyone you meet is.
Simply driving through the countryside is a great experience. It's so different from almost any other place on earth, and it's so remote that you can easily enjoy the natural surroundings from any of the main roads in the area.
Fondest memory: Again, I would have to say that my favorite part was the landscape. It's something everyone should see.