How do I insert the current filename into the contents in Vim?

The current filename is in the "% register, so you can insert it (while in insert mode) with <c-r>%; the full path can be inserted with <c-r>=expand("%:p"). You could make a macro of it if you use it often. For more, similar tricks, see :h expand and :h "=.

I'm sure there are other ways to do this... but the main command you want is :r[ead] which inserts the output of a command into the buffer.

So, to insert just the file name:

:r! echo %

And, to include the full path:

:r! echo %:p

For more info:

:help read
:help filename-modifiers 

As can be seen in :h registers, the "% register contains the current file name. The :pu[t] command inserts the content of a register into the text.

So, to insert the actual filename you can type either of these, in command mode:

:put %

or

"%p

To insert the filename with the full path, type

:put=expand('%:p')

in command mode.

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More info:

:h pu[t]

By typing "rp you can paste the contents of register "r.