I'm pretty new to Blender, but I am stuck on one thing: every time I try to move a particular vertex, or edge loop, or whatever, using the Translation Widget or the "G" key, a number of other vertices and edges move as well, as if the rest of the object is reshaping as well. However, I want to simply move the vertices or edges that I have selected, while keeping all other vertices fixed. How could I do that?
Most likely you've turned on proportional editing.
To turn it off press the O key.
Or use the proportional editing icon on the bottom of the screen:
Follow this link for more information on proportional editing