<p>If a file is set to read only mode, how do I change it to write mode and vice versa from within Emacs?</p>

M-x read-only-mode
in very old versions of Emacs, the command was:
M-x toggle-read-only
On my Windows box, that amounts to Alt-x to bring up the meta prompt and typing "read-only-mode" to call the correct elisp function.
If you are using the default keyboard bindings,
C-x C-q
(which you read aloud as "Control-X Control-Q") will have the same effect.  Remember, however, given that emacs is essentially infinitely re-configurable, your mileage may vary.
Following up from the commentary:
you should note that the writeable status of the buffer does not change the writeable permission of the file.  If you try to write out to a read only
file
, you'll see a confirmation message.  However, if you own the file, you can write out your changes
without
changing the permissions on the file.
This is very convenient if you'd like to make a quick change to a file without having to go through the multiple steps of add write permission, write out changes, remove write permission.  I tend to forget that last step, leaving potentially critical files open for accidental changes later on.