I'm old school. I do a lot of warm up drills that I learned back in the late 70's early 80's that I did before football or wrestling. I start at the top and work my way down. I will start with several neck stretches, arm rotations, chest flies, side bends, trunk rotations (or torso twists), lunges, body weight squats, and calve raises. Sometimes I may incorporate more but this is basically it.
I also may try a light set of a particular exercise but this is not for stretching but if I have been away from a particular exercise so that I can feel the movement and can feel the muscles that I need to concentrate on. Otherwise I find that after my originally stretching and doing the 5 sets that I'm more then warmed up.
Now cooling down is another thing altogether which I need to work more on ... and something that many people forget. If you are raising your heart rate you also have to lower it. As you get older you have to make sure you just don't stop as that is not good for the old ticker. Walking around until you get it under control. Do something light like stretches. This should be an important addition to anybody's workout routine.