How to switch to a different buffer from a terminal buffer

In term-mode, any regular C-x whatever keybinding becomes C-c whatever instead.

I'm not sure I understand your question. If you run M-x terminal, most of the key events are sent to the underlying terminal, so the standard C-x o binding and your M-Left are not available in the terminal.

Try using M-x shell to get a shell in one of the windows, and the navigation bindings you set up should still work.

In term-mode, type C-c b RET to switch to some other buffer.

That does what C-x b RET normally does.

This should do the trick to get C-x b working. You may have to add bindings for any custom move commands.

(add-hook 'term-mode-hook
   (lambda ()
     ;; C-x is the prefix command, rather than C-c
     (term-set-escape-char ?\C-x)
     (define-key term-raw-map "\M-y" 'yank-pop)
     (define-key term-raw-map "\M-w" 'kill-ring-save)))

BTW, there is a big difference between shell-mode and term-mode. The former integrates better with emacs (e.g. cd command). The latter is a full terminal emulation and can handle curses programs. They both have their place.

For a more generic answer dealing with emacs' windows, you can look at windmove, which started shipping with Emacs circa Emacs 22, I believe:

;;; Commentary:
;;
;; This package defines a set of routines, windmove-{left,up,right,
;; down}, for selection of windows in a frame geometrically.  For
;; example, `windmove-right' selects the window immediately to the
;; right of the currently-selected one.  This functionality is similar
;; to the window-selection controls of the BRIEF editor of yore.
;;
;; One subtle point is what happens when the window to the right has
;; been split vertically; for example, consider a call to
;; `windmove-right' in this setup:
;;
;;                    -------------
;;                    |      | A  |
;;                    |      |    |
;;                    |      |-----
;;                    | *    |    |    (* is point in the currently
;;                    |      | B  |     selected window)
;;                    |      |    |
;;                    -------------
;;
;; There are (at least) three reasonable things to do:
;; (1) Always move to the window to the right of the top edge of the
;;     selected window; in this case, this policy selects A.
;; (2) Always move to the window to the right of the bottom edge of
;;     the selected window; in this case, this policy selects B.
;; (3) Move to the window to the right of point in the selected
;;     window.  This may select either A or B, depending on the
;;     position of point; in the illustrated example, it would select
;;     B.
;;
;; Similar issues arise for all the movement functions.  Windmove
;; resolves this problem by allowing the user to specify behavior
;; through a prefix argument.  The cases are thus:
;; * if no argument is given to the movement functions, or the
;;   argument given is zero, movement is relative to point;
;; * if a positive argument is given, movement is relative to the top
;;   or left edge of the selected window, depending on whether the
;;   movement is to be horizontal or vertical;
;; * if a negative argument is given, movement is relative to the
;;   bottom or right edge of the selected window, depending on whether
;;   the movement is to be horizontal or vertical.