I don't know what kind of running regime you have tried previously but, obviously, stopping a few month is not a good idea! ;o) Being a former smoker probably doesn't have much to do with is as many might think; sure it would probably make it wee bit harder in the beginning but, as long as you have stopped it completely (hopefully), you are just at the same starting line as anybody else. I've seen people starting to run with only one lung (technically, there are 5 sacks and I think she had removed 2 of them) due to lung cancer and ended up completing a marathon in under 5 hours. Lungs are one of the most efficient organs in our body; it actually take in something like 5 times oxygen than the body can actually use! In other words, you are probably getting plenty of oxygen into your lungs. The problem is circulation of oxygen and utilization. Weezing, or heavy breathing, is actually the feedback of your musculs. Your working muscles, mainly your legs, are not toned enough to utilize enough oxygen required to complete the task (in your case, running 1/4 mile or a mile or whatever at such-and-such pace); that is your problem.
Some people might even insinuate oxygen has got nothing to do with performance and power is all you need. However, having done some strength training has very little to do with your endurance (might help some, but not much). We have what we call capillaries surrounding our muscles. These are network of tiny blood vessels around your muscles and this is where working muscles get oxygen and nutrients. By doing more exercise you do, your body would send a message to retain more oxygen and you develop vast network of these capillaries. Along the way, as you already know, your heart will become stronger and starts to pump more blood per minute. Your blood tone will also improve, enabling you to carry more oxygen to where it's needed. The redness of your blood is due to hemoglobin, a tiny iron compound, that carries oxygen. This is why, if you're anemic, low in iron, it would be hard to carry on physical tasks--you'll get tired easily. Now as you continue to exercise, you'll get better blood tone, your heart becomes bigger and stronger to send all those hemoglobin, carrying oxygen, to the working muscles, and, with good network of capillaries, muscles will use more oxygen efficiently; so now you can complete the task more efficiently.
When you do strength training, your muscle fibers increase its size (=to become stronger). But that does very little to develop capillaries. So in other words, now you have these bigger muscles with the same capillaries; the actual fact is, you're getting less oxygen to your working muscles! So this could actually defeat the purpose--there might be a time strength training CAN help; but not innitially. This is also a reason, in a way, why losing weight can help you run further and faster--you're using up more oxygen per kilogram, or pound, of your body weight.
In the beginning, it might pay to do interval of jogging and walking. Any kind of publicized beginner's program, something like C25K, should be good. Your initial goal should be to continue the exercise for 15 minutes. So you can set out to walk 3 minutes and jog 2 minutes and repeat that for 3 times; there's your 15 minutes...something like that. Some literature might suggest doing some cross training, but let's stick to running for now. Your goal should be to do this at least 3 times a week. One of the Golden Rules should be; do little often than do too much once and hang it up for some time. You need to do what you can, regardless of how little it may be, and keep doing it.
Once you feel comfortable with this regime, do it more frequently instead of lengthening it or making it harder. In your case, without knowing much about you, I think it is important to do it more often. Maybe shoot for doing this 5 days a week...something like that. Don't take 2 months break from one workout to the next! ;o) Keep on keeping on. Do little often than skip many days in between.
After awhile, 3 minutes walk--2 minutes jog regime will be too easy for you. Start lengthening your jogging segment. Start out with now 2 minutes walk and 3 minutes jog; then 1 minutes walk and 4 minutes jog... Walking breaks have its place but let's try to do a continuous jogging first. Your next goal should be to jog for 15 minutes without stopping. Once you reach this level, and, believe me, as long as you keep at it and don't push yourself too hard (in other words, don't do anything stupid!), you'll be amazed how quickly you improve. No other activity can show you such dramatic improvement so quickly. Trust me; keep at it, keep within your limitations (don't push it too hard), and keep doing it little by little; you'll see a big leap very very quickly.
Once you make it to 15 minutes, every third day or thereabout, go a little bit further; like 20 minutes, then 30 minutes; then 40... But always go back to 15 for recovery. Take some walking breaks if necessary (when going far); but try to stay on your feet for a long period of time. Once you get this far; you're on your way for whatever you might have dreamed of doing (running wise, that is). Big M or A little big HM??? ;o)
Just a small anecdote; this is exactly how Arthur Lydiard explained and instructed the first ever organized jogging group in Auckland, New Zealand, back in 1961. He had a group of 20 people, the youngest was 50 and the oldest was 74. Previously, they ALL had had at least one heart attack. None of them, in the beginning, could run 200m (a half a lap around the local track) without stopping. By doing what I've just explained, ALL of them, after 8 months, were running 20 miles without stopping and 8 of them completed the marathon. It is all possible and it is all real!
Good luck; keep at it, and happy running! ;o)
PS: I really HATE this format! Whoever comes up with this program SUCKS! I can't believe nobody has complained (I guess me included), or if we did, nobody has done anything about it... This is a typical case of, "We have a problem, but we're not going to do anythnig about it..."