Daff-
You have it right. And you are also correct, bikers pay very close attention to heart rate- but sometimes runners do, too. (I own a polar monitor, used it for both).
The thing I tell people about running is it isn't like you think it is, or should be. The worst mistake most people make is going out with the mindset it's supposed to hurt all the time. What you should do is run no faster than you can if you were trying to hold a conversation with a person running next to you. This means you are actually finding a pace where you are quite comfortable. If you get serious later on and want to train hard, you would still find that most runs in a week are comfortable. There would be only 2 or 3 days a week where the running would get uncomfortable.
Daff is right, going beyond your body's ability to provide oxygen is key (not to mention lactic acid buildup), but you also have to consider what distance you tried to start at, your weight, the type of shoes you're wearing, etc... If you are seriously competitive, you have to learn how to break your training up into phases. You have to plan long-term, and chart yourself a path to get there. One little-known thing about running is it is notoriously easy to get hurt, and taking off without a plan is a good way to do it.
Fdiron, how did you get hurt? Curious...