A number of things come to mind when I look at a coaster, but mostly I like to get into the station area and look at all the things on the track; if it's a launched coaster, I'll look at the launch too. Brakes also interest me, especially mag brakes. I think it's something to do with the way they work so flawlessly; a bunch of parts working together, timed perfectly, manipulating the behavior of the train (in some cases, the thrill it gives riders).
Other than the technical stuff, I really love looking at the shaping of the track. If a coaster does really complex maneuvers, the shape of the track at those points usually makes me like to just look at the track. Take Alpengeist for instance, I love looking at the 0g roll and the way it goes in and out of the wingover right after it.
This is one of my favorite pictures of that ride; not only is it really up close, but it gives a great view of the 0g.