Welcome to the team Ashley.
A few things...firstly, it is not too unusual to be breathless when you begin a running program. The reason, many of us run much faster than we need to so that we start going ananerobic (without the presence of oxygen) and because of this, we can only run so far before we feel the need to stop or slow down. This is one reasn why as a new runner, many programs recommend a run/walk training routine. Slowing your pace will help immensely.
Secondly, breathing through your mouth allows for more oxygen to reach the lungs which is why doing a combination nose/mouth breathing technique allows for better oxygen consumption.
Lastly, you want to practice belly breathing...the best time to do this is not when you are running but when you are lying down. The best time to do this is before bed or while watching TV. What you will want to do is place a book on your lower abdomen and practice raising and lowering the book with deep breaths.
I hope this helps!