Bicycle riding is essentially aerobic exercise. Aerobic, means it uses air. You need to intake oxygen to "burn" and expel the waste carbon dioxide produce by that "combustion". The harder you work the harder you must breath. Basically, breathing is enabling you to go up the hill, not preventing it or slowing you down.

By "getting into shape" you are conditioning your body mainly to be able to produce higher consistent power output, but also to be somewhat more efficient. But you will still need to exert yourself and breath heavily to get up hills or ride fast. As Greg LeMond purportedly said, "It doesn't get easier, you just get faster."