heart rate monitor for biking indoors and outside

What's your budget, and what features do you need? I'd recommend a combination bike computer and GPS if your budget stretches to that, so that you have one device that combines HRM and bike computer, to record where you cycled, how fast you were going, etc. They tend to be more accurate at estimating calories, too, since they take into account your weight, your bike's weight, your speed, and the hilliness of the terrain as well as your heart rate. The Garmin Edge 500 or 510 is good if you don't need navigation; the Edge 810 is better if you want navigation.