If you're using QGIS 2.14 or later you'll be able to use the handy Trace Digitizing tool we added to the Advanced Digitizing toolbar.
Follow these steps to use the Trace Digitizing tool.
For tracing to occur, you'll need to snap a vertex of your new line or polygon feature to an existing vertex or segment of one of the features you're tracing.
Set your snapping parameters via Settings > Snapping options ... from the QGIS main menu.
Enable editing for your line or polygon layer. If you look closely at the Advanced Digitizing toolbar, you'll notice a new icon:
Note: If you can't see the Advanced Digitizing toolbar, you can enable it via View > Toolbars > Advanced Digitizing Toolbar from the QGIS main menu.
The tool can be enabled/disabled by clicking the icon. You can also toggle the tool by pressing T key on your keyboard. This shortcut key is very useful when you're in the process of digitising.
To use the tool, simply enable Trace Digitizing and click on / snap to a vertex or segment of an existing feature you wish to trace along.
You can now hover the mouse pointer along various features and the red rubber band will indicate the proposed trace path from the last point you snapped to the traced layer. Left-click and you should have now captured the whole length of the path in one click!
Note: If there are multiple paths between first point and the next point, the tool will pick the shortest path. If you want a different route, you need to click on an intermediate point.
Note: If the second point you click does not snap to a vertex or segment on the traced layer, a straight line from the start point to the second point will be captured.
The following video shows the Trace Digitizing tool in action:
You can also use the Advanced Digitizing tools (e.g. Reshape Features) in trace mode.
This video shows how you can make use of tracing while reshaping a polygon: