With the line tool (which you can access by pressing U), you can draw lines. How can I precisely adjust the ends of the line? I can't drag from the line ends (that just makes a new line), and trying to use the normal mouse tool doesn't seem to work either.
Try to use the 'Pen Tool' P and draw paths instead. Paths are the true point-to-point lines and what the 'Line Tool' does is actually a "line" with 4 corners. Then, select different points on a path using the 'Direct Selection' tool A as @AAGD already mentioned.
The 'Direct Selection' tool A also works with lines created with the 'Line Tool', but as you start dragging you will notice the line will 'transform' into a shape with 4 corners, which is actually not a line.
So P and A are the tools you should be looking at.
Editing a line drawn with the Line Tool (U) is a bit tricky, as it's really a narrow rectangle.
You can edit the end points with the Direct Selection Tool (A) but you will find that there are actually two points at each end.
This means that by moving these points around, you might actually get a thinner line, because the rectangle gets distorted.
This works for vertical or horizontal lines. It is especially useful if you need to go back in and edit lines at a later time. Draw the lines a bit longer than you need. Select a small section of the straight part of the line using the rectangular select tool and delete it. Then select the rest of the line starting from the blank area and include the entire end of the line. Switch to the move tool and then use the up and down or the left and right arrows on your keyboard to exactly place the tip of the line where you need it. When you deselect they will merge. This will keep the line width and endpoint size consistent with the original.