Maybe you are starting out running at too fast of a pace? Try starting slower then building speed if you feel able. As Beachdog said, don't concentrate on how you are breathing, your body will react naturally. Some people try to stick to the adage: "Breath in through the nose, out through the mouth." don't do that, as you work harder and the body is laboring you will need to breath in and out through your mouth to get enough oxygen.
Everyone is different and progresses at different levels. Because you are already a fit and active person, you may have had a higher expectation going into a run, but it just takes time, so be patient it will come
When I began to train for my first triathlon I was a seasoned runner and considered myself to be in good shape....that is until I began swimming! I swam one 50 yd up and back length of the pool and thought my lungs and heart were going to burst out of my chest!!
I realized then that different sports/activities require different conditioning even for a fit person.
Lastly, in answer to your treadmill versus street question...I'm not sure which is better for a more productive workout, they are both good, I personally prefer street or trail running, it just is more fun and interesting, a treadmill is definitely my second choice.
Good luck to ya!!