I have been to Big Bend twice; once by myself and once with my then wife-to-be. In both cases, my first hike was the Lost Mine Trail. It is a really pretty hike of moderate difficulty that gives a good overview of the flora and geology of the Chisos mountains. The self-guided nature of the trail allows you to recognize the various plants and geological features you will encounter on numerous other hikes in the Chisos and even in the desert sections of the park.
I was particularly impressed and took a picture at the overlook of the ridge which shows the difference in plant life between north and south facing slopes in Big Bend.
PS: I saw a black bear one evening near the Lost Mine Trailhead.