I had the same problem too at first. I was mashing around in too high of gears and eventually you reach a point where you're pushing too hard but your heart and lungs can't keep up. I got a cycling computer that measures cadence and I started spinning faster in lower gears to build up my cardiovascular system. At first, my cadence was typically between 70-80 and now it's usually between 90-100 and I can push higher gears and go a lot faster than I used to. I rarely feel tired the next day.

At first when you start spinning more and mashing less, you feel ridiculous and out of control. But you adapt quickly and now it annoys me when my cadence falls below 80 - I shift down and spin more.