Hi, Helix!
Congratulations for having the sense to get back into shape while you're still young! I waited until I was 59 and 120 lbs overweight to make that decision, and it's a lot harder now than it would have been at your age, lol.
I also have asthma, and I can tell you what helps me. Of course, every body is different, and yours may react differently to the 'stress' of exercise. It may react worse, depending on the severity of your asthma, or your youth may help you out in that area. But for me, the key is to start slow and easy, and to progress slowly. I know the temptation is to see how fast you can get up to X miles or X speed, but for asthmatics that can be a dangerous temptation. I started out very slowly at very low mileage, both because of my asthma and because I was so old and out of shape -- so I only did about 1 mile at 5-6 mph my first day. Every time I try to make a jump in mileage or distance too rapidly, my asthma flares, so I've had to learn not to do that. I try to progress at a rate of about 10% per week. So if I did 10 miles one week, I did 11 miles the next, not 30. It's a little frustrating to go so slowly, but I'm now up to 38 miles a day, and my asthma hasn't acted up lately. I AM very careful, when I'm riding, to pay attention to my breathing and to keep it deep and regular, instead of panting.
Another thing that helps me is to watch my allergies. Since asthma is a form of allergic reaction, it acts up in me whenever my allergies are acting up. So I take allergy medicine almost preventatively, and my asthma has been much better since I started that.
I hope this helps. Keep your inhaler with you, and let us know how it goes. Good luck!