Really it's no different than any other type of working out; the more you do it, the better you'll get at it. Running is a bit of an extreme though. You will build it up relatively quickly, but you'll also lose it fast if you don't keep it up.
I've never been a runner, but last year I wanted to the do the City-Bay Run (12km) finally after putting it off for years. 5 weeks before I started jogging on the treadmill twice a week. Just 30 minutes at a time. I would run as far as I could, slow it down to a walk for a break, and then finish the 30 minutes. Every week I tried to hold off on the break for an extra minute or 2. After repeating this for 5 weeks, I was about to run for 40 minutes without a break. However this was still only half the distance of the City-Bay Run. However on game day, I was able to dig deep and complete the 12kms with no breaks. Being out in the open and competing against others gives you extra drive you don't have on a treadmill.
Actually, the treadmill can be the best and worst thing about running. It's great at controlling your progress and being able to exactly monitor it. But it's incredibly boring and makes working out a chore. I found when training, 75% of the challenge was in my mind. For example, you jump on the treadmill, get up to speed and you're planning to run for 30 minutes. After 2-3 minutes you're already breathing heavy and working up a sweet, then you try to figure out how you can keep it up for the next 27 minutes. At the time it seems impossible. However after about 10 minutes you get into a groove and can tune everything out.
So if you're struggling with running without breaks, grit your teeth and do at least 10 minutes. After this amount of time you seem to relax into it.
As for general tips; get good shoes, and good socks while you're at it. I struggled with blisters for a while and after researching a bit, socks play a big part in your skin rubbing against your shoe. Get nice thick soft socks.