Breathe more deeply?
It could be that two miles isn't enough to get really warmed up, especially if it's uphill to start with. The body takes time to get into the swing of it. One training strategy could be to start the ride at a speed you know can maintain without stopping - make 'not stopping' the goal, no matter how slowly you're going. And time yourself to the store. Once you've established that base 'non-stop' time, you can try to gradually bring the time down little by little. Could be fun!
On breathlessness specifically, when I feel out of breath, often at the start of a ride or on the first hill, I start breathing really deeply and in rhythm with my peddling, so I'm taking in as much air as I possibly can - even more than I need, or so it feels like to start with. But I find it helps in getting up the hills, and getting through that nasty shortness of breath feeling. Sort of like 'In, In, IN! Out, out, out', or whatever rhythm works for your peddling speed. Just make sure to REALLY fill your lungs on the last intake before exhaling, and do it all rhythmically - 2 in & 2 out, or 3 in & 3 out, or 4, or whatever. Makes a big difference I find. Most of us breathe very shallowly most of the day. It's surprising how much more air the lungs can actually hold if we fill them to capacity, and one of the best remedies for breathlessness is to take in more air... :-) It will also increase your lung capacity over time - a good thing. Just consciously taking in more air in a rhythmic and focussed manner could improve your performance a lot.
Just keep at it, and maybe look for opportunities to ride a bit more often. You'll get better. Muscles are just muscles, including the heart -and lungs up to a point. If you're not riding a lot, maybe look for some other aerobic activity on the off days - even just fast walking. Or go to the store 3 or 4 times a week instead of 2, or look for other rides to do on other days. Integrating the bike into your life is the best and most painless way to get better at it, and get more fit.