Is working your entire body a good way to put together an exercise program?

Absolutely, using the entire body requires you to use the core muscles to stabilize your body, so why not whittle away at the waist at the same time. Because the core muscles have to transfer the force between the upper and lower body, it requires more muscle activation and thus a higher caloric expenditure. You can get more done in less amount of time!

If you'd like a progression from the circuit machine, try this exercise at a cable cross machine.

The Squat to a Row is a total body exercise. When you sit in a machine- it provides an external support, so your internal support (the abdominal muscles) doesn’t have to work as much.

1. Attach a straight arm bar to the cable.
2. Extend arms forward as you lower to a 90 degree squat.
3. As you stand up, pull the bar in to mid-chest.
4. Set weight to reach muscle fatigue in 2-3 sets of 12-15 repetitions. 

Because this is a multi-muscle movement exercise, you could reduce the weight, increase the reps and use it as a warm-up, or as cardio circuits in-between your strength training exercises.

It is a great way to workout out.  Working out multiple muscles will help you burn more calories than you are use to.  This is also a great way for beginners to start an exercise program so they can get use to what they need to do.  As you progress you can change your program as you see fit to help you reach your goals as your body adapts.