I need to create a user comment form with three text areas and then display them in comments. I don't know where to start - cannot find any plugin but would like to code it anyway. Any pointers much appreciated.
To add new fields, simply append them to the default fields with a plugin.
Then there's also the possibility to use comment meta data. More about it can be read in the Codex.
To attach the meta data, you can hook into an action:
Note: This is not tested, but should give you a starting point.
3 textares for what purpose? or you meant 2 input´s and 1 texarea plus 1 input button? i would start reading at http://codex.wordpress.org/Function_Reference/comment_form