I think you're more likely to bounce when the seat is too low, if that's the culprit, like OakLeaf said. It's because you bottom out on the pedal stroke too early. That doesn't really sound like the issue here, but you went in too easy of a gear for you to be able to pedal smoothly. Being able to spin up really fast requires practice. I find it's easier to practice doing that downhill when you start running out of gears, because whether you can pedal smoothly or coast doesn't matter like it does uphill. So, it sounds like you did the right thing in shifting slightly harder. Because it was harder, you felt your muscles burning. That doesn't mean you still weren't "spinning." The way my musculature is naturally, I tend to feel more comfortable at a lower cadence on hills, but like many, I quickly found out that my legs would be too tired at the top, and my joints would eventually complain. So for me, learning to "spin" meant dealing with some serious muscle burn on longer climbs. The way to improve is to deal with that as long as you can, by trying to keep a smooth rhythm and breathing as calmly as possible. Then shift one way or the other if you need to "rest" or change how you enlist your muscles if you haven't made it all the way up. Keep at it and you'll start to get stronger and fitter. Your pedal stroke will improve, and spinning will get easier. You will start to gravitate towards a natural gearing and cadence, but it will require experimentation and practice to figure out what "sustainable effort" means for you. Use whatever ring and whatever rear cogs it takes.