It's no coincidence that the scenery is stunning: the route is carefully planned to show France at its best.
Mountain stages are generally most interesting to me, as it thins the field and sees most attacks. You'll also see just how close the crowd get: it's fairly ridiculous at times. I do like it when there's a good climber in contention to win the GC. 2012 I found a little dull, as Sky did a very professional job in defending time in the mountains, and then winning it in the time trial. As a spectacle, I found it a bit lacking.
I do also enjoy a good bunch sprint as well mind, particularly when you see the lead out train working well. I did get a little tingle down the spine when Wiggins hit the front with 1km to go in Paris: that was just something you don't usually see.
My favourite random thing is probably the transition between climb and descent. On the way up, you'll see shirts open, water being poured over heads, anything to keep cool. As they reach the top, they often take newspaper from the crowd and stuff it under their tops as insulation for the way down. They can get up to about 60mph downhill, so there's a fair wind-chill.
My String Journey: http://tt.tennis-warehouse.com/showthread.php?t=362403