For all of us there is an optimum RPM (cadence) that minimizes damage to your knees, produces the most efficient transfer of power to the wheels, and delays the onset of fatigue. You haven't figured out what that is yet. Spinning like mad obviously doesn't do it. Neither will mashing. Don't be afraid to use your gears to find the best cadence. I replace gear cables more often than tires. Shifting will become intuitive with some experience. I automatically downshift when approaching a stop or just before reaching a hill and shift up soon after starting out or going down a hill. I also shift with changes in the terrain. My routes are rarely completely flat. Yes, you will get better the more you ride but this is one thing that can hasten your improvement.

Thanks for the tip on downshifting -- I hadn't actually thought about that. I've been reading this link Complete Guide to Bicycle Gears and Shifting for Beginner Cyclists and it's helpful, but sometimes it's hard to remember when you're moving

All of the advice I've been getting here is helpful. Last week I struggled through the ride I took today. Today I extended that ride and took a turn I don't normally take just because the grade was a little steeper and I wanted to see if I could do it. As I've said, I was slightly winded after the ride and my legs hurt -- but I still took a walk afterwards to check out the neighborhood. Then I realized how relaxed I was. And I thought "hey, that was a pretty good ride." And it was -- because it was about learning how to use the bike to do what I needed to do.

Not bragging, but I do learn pretty fast.