So I'm trying to get into better shape, who isn't? And from what I've seen, some combination of aerobic training and weight training is the best way to do it, exercise-wise. Also, from what I've read on some articles(http://www.nerdfitness.com/blog/2010/02/01/what-burns-more-calories-cardio-intervals-or-weight-training/ for example), it should be possible to do aerobic training for longer intervals of time, and more frequently, than weight training, which is possibly its main selling point as far as fitness goes. I could be wrong about that, and if I am feel free to let me know, but that's not really my question.
My question is:
I simply cannot run. The last time I tried running(more like jogging really), it went fine for the first 1/4~1/3 of a mile, then my body just sort of gave up, I started coughing up copious amounts of mucus, which eventually changed to include blood after a few minutes, and could barely breathe. Is that normal(I have very mild asthma)? That sort of thing never happens except for when I'm suffering from some respiratory illness, or am trying to run. The only "standard" I have to go on is that about 80% of the people I talk with talk about how they go on daily runs of about 2 mi, and how my school is trying to "conscript" everyone to take part in a 5 km run, and no one else seems to have this problem with it. Needless to say, 5 km and 2 mi are both well above 1700 ft. I can, however, bike, at about 11-12 mph I think. I think the longest ride I've been on biking was around 24 mi, and there was no problem there.
tl;dr: Is it normal/worrisome that I potentially cough up blood after running what seems to be a very short distance? If not, how can I remedy that? And can biking at what's apparently a pretty slow speed(at least compared to what I can find on Google) going to have the same sorts of effects on health/fitness as running can, or am I just pretty much screwed as far as getting those effects goes?
EDIT: Just a note about my biking speed, that 11.3 mph or so is from an app I have installed on my phone that acts as a pedometer designed for running and biking. When I bike, however, I do so on roads which are fairly smooth, but there is often traffic. My bike is a refurbished Raleigh from the 1950's, if I'm not mistaken, so it's very light and pretty well-constructed, but doesn't have the thinnest tires known to man. Also when I bike, there's traffic and intersections, and I'm not sure if the app calculates the waiting time for automobile traffic in its speed calculation, so I honestly don't know if my speed is remotely OK or not.