my search has turned up some references to how to change something called index.php
What you refer to is irrelevant. It's only for WordPress.ORG installs. You have a WordPress.COM blog and that does not apply.
It's not possible to achieve that on a free hosted WordPress.com blog. Every post has the publication date embedded in the URL.
Blogs are essentially a reverse chronological order publishing tool and you cannot change this order on a free hosted WordPress.com blog. Posts will be posted to your blog in accord with their datestamps at publication. WordPress by default shows posts in reverse chronological order (newest post on top). For returning visitors, this is very convenient since they can see the latest post without having to page down through page after page of hundreds of posts they have already read.
See here for the option of creating a book-like structure > http://en.support.wordpress.com/write-a-book/ Perhaps that will work for you.
You could use a sticky post as an index page > http://en.support.wordpress.com/posts/post-visibility/#sticky-posts
You may like this though Create a random post icon > http://onecoolsitebloggingtips.com/2011/02/13/wordpress-create-a-random-post-icon/