This is of course not supposed to be a comprehensive guide, just enough information to be reasonably proficient with Vim.
Everything in Vim can be configured on a user basis by editing the .vimrc file in the users home directory.
To begin typing, simply hit 'i' on your keyboard. Alternatively, you can probably press your Insert key. Pressing one of these keys puts you in insert mode. This mode is simply for entering text, hit the Escape key once you are finished typing.
The above commands are usually just abbreviated ':w' and such, there isn't really any point in typing ':write'.
When navigating the help, use Ctrl-] to follow a hyperlink and Ctrl-T to go back.
Hyperlinks are denoted by the surrounding 'bars' e.g. |bars|
If the mouse is enabled (via ":set mouse=a") you may also double-click the left mouse button on a tag between |bars|.
The commands can also be forced by appending a '!'. So ':q!' will force-quit, leaving Vim and ignoring the possibility of changes in the current file. Some can also be combined; a common combination is ':wq', quiting and saving the file at the same time.
All the above will work in either Vim or Vi and is enough to get started. The next logical step in learning Vim, though, are selections: we want to be able to cut and paste!
y 'Yank', or copy the current selection. c Cut the current selection. d Delete selection.
Indent the current selection.
< De-indent the current selection.
While in normal mode, pressing 'p' will paste.
In normal mode,
:s/(search regex)/(replace string)/
to replace text on the current line, or
:%s/regex/replace/
for the whole buffer.
Also, it can be quite useful to have multiple 'windows' open in vim to edit/view multiple files. Vim can split both vertically and horizontally. Accessing the windows functions requires using the 'ctrl-w' combination. Some of the combinations below work if you keep holding ctrl down, for example '<ctrl-w><ctrl-w>' also navigates to the next window.
There are many, many ways to be more productive in Vim. This is just a short and very basic introduction. Listed here are a few commands that may come in handy.