I think your exercise routine is adequate. But you'll always get your breathing going up a bit when you climb multiple flights of stairs. Just think about what you're doing for a second. Essentially, you're climbing like 30 foot of ladder in a minute. That's a lot of leg lifting.
Now if you're really getting COMPLETELY winded, as in huffing and puffing out-of-breath with a sweatsoaked shirt, then you need some more fitness. But I doubt that's the case. Hell, even I don't have that problem and I weigh 290 lbs. Even if it was the case, you'll need to give the treadmill a few more weeks before your lungs and heart really start to get the good benefits.