What everyone forgets is that the heart is a muscle too (got told this when my hubby had a heart attack 3 years ago from the medical staff) so just like your legs, it takes time to build up it's strength. I got the heart racing like it would explode thing when I first started cycling late last year. I've been riding regularly (commuting to work - and it's uphill all the way back) and gradually I've been getting better. It was quite hard work today because there was a fairly strong headwind - enough to make me consider getting off the bike and walking. I made it up the hill a bit slower than usual. Normally it takes me 17 minutes to get home now. Today it took 19 because of the wind. I was a bit out of breath when I came home (I'm an asthmatic too but stopped all medication about 18 months ago myself). Everything came back to normal within about 5 minutes.

For the record, since I started cycling my asthma has all but gone and I breathe much more deeply now so my lungs are strengthening too even though I still smoke.

Keep at it. Things will get better