Can we see the shape of the Milky Way while on Earth? Why or why not?
No, because we're in it.
The image of a spiraling galaxy is a beautiful thing, but can only be appreciated from outside said galaxy. Since the stars in this spiral are basically arranged on a more-or-less flat plane, the best we can see of the "big picture" is a vertical line-up of a side view.
We can however see the spiral arm that we happen to live in, provided you live in a dark part of the world where you can see that part of the Milky Way Galaxy. This gave astronomers a clue that we must be living in a spiral galaxy similar to other ones that could be seen. See pic.