To create a "curve" anchor point (also known as a Bezier point), when you click your mouse on the location of your anchor point, don't immediately let go; drag a little, until you see the connecting lines between your points begin to curve and two antennae-like "handles" appearing to either side of the point. You can use the Subselection Tool to grab these "handles" and drag them about to control the size and angle of the curves to either side of the point.
There's also a combination of the two called a "Bezier corner", which is an anchor point that has a straight line to one side and a curve to the other side. To create one of these, you need to first have an anchor point that is corner-only; a Bezier point created immediately adjacent to that one will automatically be a Bezier corner with only one handle and a curve to only one side, because one side of it connects to a corner point.
Any lines drawn, whether in a closed shape or just an open path, will be stroked with your selected stroke color.