I understand your frustration! When I started running about four years ago, I'd never been any good at running so it was something I wanted to "conquer." What got me started was reading an article about training for a 10K in 12 weeks. The article stated that even someone who hadn't run before could be ready for a 10K in that length of time. Sounded like a good challenge.--
For this program, you start out with small goals and gradually build up. The first run is about 10 minutes, then two days later run for about 12 minutes, then two or three days later run for 15 minutes, etc. Take it as slow as you need to. As you gradually increase your running time, after a few weeks, add some short periods of moderate running (ie., pushing it just a little more than you have been).
It's surprising how much of the training is mental - convincing yourself you can keep going. I wouldn't be surprised if you just pushed yourself too hard and that's what made you sick. Running is very strenuous and can really upset your system. Give yourself an hour after a meal before running, and don't drink too much right before, either.
It can be very frustrating to be around people who say they're going to try running a few miles and then just go out and do it. They're the minority though - really. Just go your own pace and build up your time/distance gradually and it will get better.