Navigation:

h  "Left
j  "Down (physical line)
k  "Up (physical line)
l  "Right
gj "Down (visual line)
gk "Up (visual line)

Moving lines:

:m+1   "Move the current line down one line
:m-2   "Move the current line up one line
:m$    "Move the current line to the end of the file
:10m20 "Move line 10 to line 20
:20m10 "Move line 20 to line 11
:m10   "Move the current line to line 10 or 11, depending on whether the current line is before or after

Turn off highlighting:

:noh

Open multiple files as tabs (from outside of vim):

$ vim -p file1.cxx file2.cxx " Globbing works, too.

Open a file as a tab (from within vim):

:tabedit file3.cxx

Navigate between tabs:

:tabn     " Next tab
:tabp     " Previous tab
:tabfirst " First tab
:tablast  " Last tab
{i}gt     " Jumps to the i'th tab

Move a tab:

:tabm 0 "Moves the current tab to the specified index

Find and replace:

:s/replaceThis/withThis/gc      " Current line
:%s/replaceThis/withThis/gc     " All lines in document
:15,20s/replaceThis/withThis/gc " Lines 15 to 20, inclusive
" :s is short for :substitute
" c requires confirmation

Jump to…

0   "Beginning of a line
$   "End of a line
:76 "Line number 76
(   "Previous sentence
)   "Next sentence
b   "Previous word
w   "Beginning of next word
e   "End of next word

Settings:

:set nu        "Turn on line numbers
:set nonu      "Turn off line numbers
:set wrap lbr "Wrap long lines between words

Syntax highlighting:

:syntax on "Turn on
:syntax off "Turn off

Alter case of selected text:

u "Lowercase
U "Uppercase
~ "Invert Case

Undo/Redo:

u "Undo
Ctrl+r "Redo

Inserting text:

i "Insert text before cursor
a "Insert text after cursor
I "Insert text before first non-whitespace character
A "Insert text at end of line

Nice things to put in .vimrc:

" Wrap long lines between words
set wrap lbr

" Add line numbers
set nu

" Highlight txx as cpp
autocmd BufReadPost,BufNewFile *.txx set filetype=cpp

Remove trailing whitespace:

:%s/\s\+$//gc " replace '\s\+$' with ''

Highlight Non-ASCII Characters:

:set hlsearch " Make sure that what you search for will be highlighted
/[^\d0-\d127] " Highlight ASCII digits (0 to 127)
:nohlsearch   " Turn highlighting back off