In general, it's true that as one becomes more fit ones resting heart rate tends to decline. Super-fit endurance athletes may have resting heart rates of 45 or 50, whereas the average is 70-75. But people start from different places -- some people have a tendency to higher heart rate or lower heart rate regardless of fitness level. And what MackTheKnife said is important, it's probably a better indicator of fitness how fast your heartbeat returns to its rest state.
Another thing to watch for with resting heart rate is that you have to be really resting, like just waking up or having been laying down for 20 minutes or so. As you're going about your day-to-day activities, even if not exercising, your heart rate will probably be above its resting rate. And just before exercising your heart may speed up in anticipation of the exercise to come.
The heart monitor is a great tool, but I'd be very wary of a fixed "fitness index".