Doug asks: "As you're painfully aware, I have just started using a Mac. I can get around my Windows computer with no problems, but the tray of icons at the bottom of the Mac screen seems awfully small. Is there some way to make it bigger?"
Doug, I feel your pain. After using Windows exclusively for nearly 25 years, I started using Mac about four years ago and remember the pains it took to learn the operating system.
You can easily show or hide the icon dock (as it's called), and you can change its size. Look carefully at the right hand side of the dock, between the main group of icons and the small set on the right. You'll see little tread marks that look like the divider down the middle of the road. If you click and drag on those tread marks, you can change the size of the dock.
If you right-click on those tread marks you'll find other options, as well. For example, you can turn hiding for the tray on or off. You can turn magnification of icons on or off. (That is, as you drag your mouse over the icons, if magnification is on then you'll see a magnified view of each icon.) You can control the dock's position on the screen and you can control other dock preferences as well.
But to answer your specific question, clicking and dragging on the tread marks will increase the size of the dock containing program icons.
As you'll find in any graphical operating system, there are a lot of non-discoverable features on the Mac and you have to use the operating system, and dig around a little, to find them. This is just one of the many features like that.