You need to do 2 different things to achieve this.
Getting the URL in your desired format: http://website.com/user/username
By default, user's archive URL is something like this http://website.com/author/username
There is a plugin to change author slug, install this plugin and set the slug to user
[Note: I'm the author of this plugin]
author.php file in theme directory and show specific information.
In this file, you can get any of information a user has.
To get a user's display-name, simply use
<?php the_author_meta('display_name' ); ?>
Or to get his first name, use
<?php the_author_meta('first_name' ); ?>