Traversing text in Insert mode

While it may make sense that you should be able to use the h j k l keys to traverse the editor in insert mode, but that is actually not the way Vim is intended to be used! There are many commands that Vim provides to make editing faster and easier.

The right way is to press Esc, go where you want to do a small correction, fix it, go back and keep editing. It is effective because Vim has much more movements than usual character forward/backward/up/down. After you learn more of them, this will happen to be more productive.

Here are a couple of use-cases:

• You accidentally typed "accifentally". No problem, the sequence EscFfrdA will correct the mistake and bring you back to where you were editing. The Ff movement will move your cursor backwards to the first encountered "f" character. Compare that with Ctrl+←→→→→DeldEnd, which does virtually the same in a casual editor, but takes more keystrokes and makes you move your hand out of the alphanumeric area of the keyboard.
• You accidentally typed "you accidentally typed", but want to correct it to "you intentionally typed". Then Esc2bcw will erase the word you want to fix and bring you to insert mode, so you can immediately retype it. To get back to editing, just press A instead of End, so you don't have to move your hand to reach the End key.
• You accidentally typed "mouse" instead of "mice". No problem - the good old Ctrl+w will delete the previous word without leaving insert mode. And it happens to be much faster to erase a small word than to fix errors within it. I'm so used to it that I had closed the browser page when I was typing this message...!
• Repetition count is largely underused. Before making a movement, you can type a number; and the movement will be repeated this number of times. For example, 15h will bring your cursor 15 characters back and 4j will move your cursor 4 lines down. Start using them and you'll get used to it soon. If you made a mistake ten characters back from your cursor, you'll find out that pressing the ← key 10 times is much slower than the iterative approach to moving the cursor. So you can instead quickly type the keys 12h (as a rough of guess how many characters back that you need to move your cursor), and immediately move forward twice with ll to quickly correct the error.

But, if you still want to do small text traversals without leaving insert mode, follow rson's advice and use Ctrl+O. Taking the first example that I mentioned above, Ctrl+OFf will move you to a previous "f" character and leave you in insert mode.

While in insert mode, use CtrlO to go to normal mode for just one command:

CTRL-O h  move cursor left 
CTRL-O l  move cursor right
CTRL-O j  move cursor down
CTRL-O k  move cursor up

which is probably the simplest way to do what you want and is easy to remember.

Other very useful control keys in insert mode:

CTRL-W    delete word to the left of cursor
CTRL-O D  delete everything to the right of cursor
CTRL-U    delete everything to the left of cursor
CTRL-H    backspace/delete
CTRL-J    insert newline (easier than reaching for the return key)
CTRL-T    indent current line
CTRL-D    un-indent current line

these will eliminate many wasteful switches back to normal mode.

Insert mode

Movement

hjkl

Notwithstanding what Pavel Shved said - that it is probably more advisable to get used to Escaping Insert mode - here is an example set of mappings for quick navigation within Insert mode:

" provide hjkl movements in Insert mode via the <Alt> modifier key
inoremap <A-h> <C-o>h
inoremap <A-j> <C-o>j
inoremap <A-k> <C-o>k
inoremap <A-l> <C-o>l

This will make Alt+h in Insert mode go one character left, Alt+j down and so on, analogously to hjkl in Normal mode.

You have to copy that code into your vimrc file to have it loaded every time you start vim (you can open that by typing :new $myvimrc starting in Normal mode).

Any Normal mode movements

Since the Alt modifier key is not mapped (to something important) by default, you can in the same fashion pull other (or all) functionality from Normal mode to Insert mode. E.g.:
Moving to the beginning of the current word with Alt+b:

inoremap <A-b> <C-o>b
inoremap <A-w> <C-o>w

(Other uses of Alt in Insert mode)

It is worth mentioning that there may be better uses for the Alt key than replicating Normal mode behaviour: e.g. here are mappings for copying from an adjacent line the portion from the current column till the end of the line:

" Insert the rest of the line below the cursor.
" Mnemonic: Elevate characters from below line
inoremap <A-e> 
    \<Esc>
    \jl
        \y$
    \hk
        \p
        \a
" Insert the rest of the line above the cursor.
" Mnemonic: Y depicts a funnel, through which the above line's characters pour onto the current line.
inoremap <A-y> 
    \<Esc>
    \kl
        \y$
    \hj
        \p
        \a

(I used \ line continuation and indentation to increase clarity. The commands are interpreted as if written on a single line.)

Built-in hotkeys for editing

CTRL-H   delete the character  in front of the cursor (same as <Backspace>)
CTRL-W   delete the word       in front of the cursor
CTRL-U   delete all characters in front of the cursor (influenced by the 'backspace' option)

_{(There are no notable built-in hotkeys for movement in Insert mode.)}

Reference: :help insert-index

Command-line mode

This set of mappings makes the upper Alt+hjkl movements available in the Command-line:

" provide hjkl movements in Command-line mode via the <Alt> modifier key
cnoremap <A-h> <Left>
cnoremap <A-j> <Down>
cnoremap <A-k> <Up>
cnoremap <A-l> <Right>

Alternatively, these mappings add the movements both to Insert mode and Command-line mode in one go:

" provide hjkl movements in Insert mode and Command-line mode via the <Alt> modifier key
noremap! <A-h> <Left>
noremap! <A-j> <Down>
noremap! <A-k> <Up>
noremap! <A-l> <Right>

The mapping commands for pulling Normal mode commands to Command-line mode look a bit different from the Insert mode mapping commands (because Command-line mode lacks Insert mode's Ctrl+O):

" Normal mode command(s) go… --v <-- here
cnoremap <expr> <A-h> &cedit. 'h' .'<C-c>'
cnoremap <expr> <A-j> &cedit. 'j' .'<C-c>'
cnoremap <expr> <A-k> &cedit. 'k' .'<C-c>'
cnoremap <expr> <A-l> &cedit. 'l' .'<C-c>'

cnoremap <expr> <A-b> &cedit. 'b' .'<C-c>'
cnoremap <expr> <A-w> &cedit. 'w' .'<C-c>'

Built-in hotkeys for movement and editing

CTRL-B       cursor to beginning of command-line
CTRL-E       cursor to end       of command-line

CTRL-F       opens the command-line window (unless a different key is specified in 'cedit')

CTRL-H       delete the character  in front of the cursor (same as <Backspace>)
CTRL-W       delete the word       in front of the cursor
CTRL-U       delete all characters in front of the cursor

CTRL-P       recall previous command-line from history (that matches pattern in front of the cursor)
CTRL-N       recall next     command-line from history (that matches pattern in front of the cursor)
<Up>         recall previous command-line from history (that matches pattern in front of the cursor)
<Down>       recall next     command-line from history (that matches pattern in front of the cursor)
<S-Up>       recall previous command-line from history
<S-Down>     recall next     command-line from history
<PageUp>     recall previous command-line from history
<PageDown>   recall next     command-line from history

<S-Left>     cursor one word left
<C-Left>     cursor one word left
<S-Right>    cursor one word right
<C-Right>    cursor one word right

<LeftMouse>  cursor at mouse click

Reference: :help ex-edit-index

If you are a vim purist, skip reading this answer. OTOH, if you are new to vim and are looking for a few helpful tips you wont find in the many hundred of vim tutorials and blogs, read on... :-)

A few un-orthodox (vim) ways

It's 2014, and as someone who's recently gone back to vim I can offer a few, potentially contrarian, points of view and tips.

Use shift+left or shift+right to traverse words

While repetition is a powerful concept in vim, I (personally) find it strange that using it either forces me to count (lines, characters, words, etc.) or make guesses. My brain usually works like "I want the cursor there" and not like "I want the cursor _5_words_to_the_left_". Quickly being able to move the cursor, and visually observe where the insertion point this way allows me to keep my mind on what I'm editing instead of having to count how many hops I need to make to get to where I need to edit.

Turn on mouse mode, and use the mouse wheel and clicking

...to navigate large bodies of text.

Most (all) modern computers have a touchpad that is closely integrated with the keyboard (e.g. MacBooks). Industrial designers have spent many man years optimizing these designs so that the old problem of having to move the hand away from the keyboard is no longer a real issue. Okay, it is if you are used to a mouse and don't like to switch, but for anyone new to vim (like those that might find this post via a search), this should not be much of an issue.

As a bonus, click + drag puts you in visual mode

With mouse enabled, clicking and dragging has the effect of switching to visual mode and marking a region for yanking.

And use the scroll wheel

Using the mouse (wheel) to scroll around, and clicking to position the cursor (duh) just works. See http://usevim.com/2012/05/16/mouse/ for more on this.

And so...

These are what I'd call more modern (using mouse, scroll wheel, etc.) ways of navigating in vim, equally effective depending on your preference of input.

HTH