I was just like you when I started last year - good thing I was on back country roads, 'cuz if anyone had heard me breathing, they would have called 911. There's not a drop of exaggeration there, either. My lungs still conk out on some hills, but they have gotten much stronger over time. Also, lungs warm up same as muscles. My breathing always improves after about 20 minutes. Your rides are short now, so you don't get the benefit of the body "opening up" as much as you will later. And don't worry, your miles will increase, sure as shootin'.

Something that also used to help ( not needed too much now) as I face an incline soon no matter which way I turn out of my driveway- a trainer suggested I take a short brisk walk to help warm up legs and lungs.

Spin classes helped me a lot with breath control, though I can't bear to do them April - October, when I can be outdoors. In spin, though, I can focus on breathing - no worries about road hazards! - and work on breathing through the nose, not panting, etc. If you work with a heart-rate monitor, it's almost like biofeedback!