When I try to select all the vertices and use Snap with vertex snapping enabled, then choose the X axis, the selected vertex behaves unpredictably when I hover over the target vertex I want to snap to. Could you please help me with this?
What do you mean by "select all the vertices"?
You only want to select the vertices you wan to move, not the vertex you want to snap to and certainly not all vertices.
With snapping enabled, vertex selected in the snap target options and the correct vertices selected, left click and hold on the red X axis on the gizmo and then move the mouse over the vertex you want to snap to (align with).
A square indicator should appear over the 'snap to' vertex and the selection snap to the correct position.
Also remember before the Fundamentals there is the Blender Basics, it is the course JL always refers to as "what we saw in the basics". It is the very first step to take. So you may want to start with this course before you tackle the Fundamentals. Linking up:
https://cgcookie.com/courses/blender-basics-an-introduction-to-blender-4-x