Running/stamina question

I am going to start running again - ran on Saturday AM, then went for a hike (moderate effort) Saturday PM, then hike again Sunday (mild effort) PM, and a walk on the beach Sunday (mild effort) late PM. I would like to think I am fairly fit, I ride my horses 5-6x a week each, usually 30-55 minutes per horse (often I am more tired/sweaty then they are - good workout), plus chores around the barn.
I guess what I really want to know is what will help me build stamina the quickest for my jogging/running. When I run, it is not my legs that get tired, its my lungs that start to hurt after a very short period of moderate effort running.
Am I better off to run slower for slightly longer periods, or sprint for short periods to improve my stamina? I would like to eventually become a "runner", and ultimately do a 5K.
Any advice is appreciated :)

I found that running (jogging, really) until I was out of breath and then walking while I recovered, then running again (repeat the on/off from walking/running) was beneficial until I gradually became able to run constantly.  I'm not sure what works for others or if there is some 'fast-track' to cardiovascular stamina, but I feel that it is something you just need to work at gradually and it will improve.  Good luck!

Run as hard as you can for 800 meters - then walk for 300 - repeat five times per session to start.  One week later do it 6 times.  One week later do it 7 times.  Then run 1000 as fast as you can - then walk 300 - repeat five times per session for the first week - 6 times the next - seven the next - THEN run 1200 then walk 300 - you get the drift.

 Check out Learn To run - Again - sticked at the top of the forum. Following a walk/run plan like that (or the couch25k plan) will get you the maximum result in minimum time.

 You could do HIIT to get a higher lactic acid threshold but that only improves short-term performance; following a more sensible training progression like LTR-A gives you a solid base of muscular adaptations that are more long-lasting.

 (Of course horse riding is a workout. Anyone who doesn't think it is, hasn't thought through the implications of using your muscles to control an animal that weights ten times what you do,most of it muscle ;)

Hiah, as the others have said walking 3 mins then running 1 min for one week then gradually adding on each week definitely helps, or try it with distance as legaleli wrote. I did it a few months ago when my lungs wouldn't play ball and it did improve with this method. Even managed the 5k run and looking forward to the 10k next! Good luck and keep at it!

Besides what Melkor said (I am a big believer in following a plan, somebody elses), simply increase the length of your long run each week by 10% (time OR distance).  The best way to build stamina is time on your feet.  How fast you run does not matter.