I am trying to build my first world, and am confused about road placement. The road move/rotate tool seems to work randomly. Sometimes I am able to shape roads the way I want, but most of the times nothing happens when I click on the blue squares. Has anyone else had this issue? Or am I doing something wrong?

Are you trying to rotate an intersection, or bend a road? Could you maybe show a screenshot?

I'm trying to make a road curve by dragging the blue squares. It only works sometimes. With sidewalks though it doesn't work at all - once I place them I cannot bend them,

The roads are like a spline- you have to drag the point handles (the blue squares) to move the curve, and change its length to alter the shape. So, get the start and end points where you want them and then make the curve with the handles.
Rather like this:

Thank you. I think I'm beginning to understand what's happening. When I place a single stretch of road I'm able to bend it (screenshot A.), but as soon as I connect it to other roads, I cannot move any parts of them at all (screenshot B.).

That sounds right, because once you join the roads, you have fixed the rotation of the end point. So you need to disconnect it to change the curve, then reconnect again (but the road shape will change if necessary to accomodate it connecting to the other road).

Ah ok, I get it now. Thank you!

No problem! CAW is definitely tricky, but once you get the hang of it, you can do pretty much anything you can imagine.