Cursor movement

• h - move left
• j - move down
• k - move up
• l - move right
• w - jump by start of words (punctuation considered words)
• W - jump by words (spaces separate words)
• e - jump to end of words (punctuation considered words)
• E - jump to end of words (no punctuation)
• b - jump backward by words (punctuation considered words)
• B - jump backward by words (no punctuation)
• 0 - (zero) start of line
• ^ - first non-blank character of line
• $ - end of line
• G - Go To command (prefix with number - 5G goes to line 5)

Note: Prefix a cursor movement command with a number to repeat it. For example, 4j moves down 4 lines.

Insert Mode - Inserting/Appending text

• i - start insert mode at cursor
• I - insert at the beginning of the line
• a - append after the cursor
• A - append at the end of the line
• o - open (append) blank line below current line (no need to press return)
• O - open blank line above current line
• ea - append at end of word
• Esc - exit insert mode

Editing

• r - replace a single character (does not use insert mode)
• J - join line below to the current one
• cc - change (replace) an entire line
• cw - change (replace) to the end of word
• c$ - change (replace) to the end of line
• s - delete character at cursor and subsitute text
• S - delete line at cursor and substitute text (same as cc)
• xp - transpose two letters (delete and paste, technically)
• u - undo
• . - repeat last command

Marking text (visual mode)

• v - start visual mode, mark lines, then do command (such as y-yank)
• V - start Linewise visual mode
• o - move to other end of marked area
• Ctrl+v - start visual block mode
• O - move to Other corner of block
• aw - mark a word
• ab - a () block (with braces)
• aB - a {} block (with brackets)
• ib - inner () block
• iB - inner {} block
• Esc - exit visual mode

Visual commands

• > - shift right
• < - shift left
• y - yank (copy) marked text
• d - delete marked text
• ~ - switch case

Cut and Paste

• yy - yank (copy) a line
• 2yy - yank 2 lines
• yw - yank word
• y$ - yank to end of line
• p - put (paste) the clipboard after cursor
• P - put (paste) before cursor
• dd - delete (cut) a line
• dw - delete (cut) the current word
• x - delete (cut) current character

Exiting

• :w - write (save) the file, but don't exit
• :wq - write (save) and quit
• :q - quit (fails if anything has changed)
• :q! - quit and throw away changes

Search/Replace

• /pattern - search for pattern
• ?pattern - search backward for pattern
• n - repeat search in same direction
• N - repeat search in opposite direction
• :%s/old/new/g - replace all old with new throughout file
• :%s/old/new/gc - replace all old with new throughout file with confirmations

Working with multiple files

• :e filename - Edit a file in a new buffer
• :bnext (or :bn) - go to next buffer
• :bprev (of :bp) - go to previous buffer
• :bd - delete a buffer (close a file)
• :sp filename - Open a file in a new buffer and split window
• ctrl+ws - Split windows
• ctrl+ww - switch between windows
• ctrl+wq - Quit a window
• ctrl+wv - Split windows vertically

Another good vim commands cheatsheet and a vi introduction using the "cheat sheet" method