Okay, let's clear a few things up. For extensive information on how to use the
<!--more--> see: Linking_Posts_Pages_and_Categories.
When used with
the_content() template tag, where you put the MORE tag (from the Quicktags above the Write Post textarea where you write your post), an excerpt will be show to that point.
If you use
the_excerpt() instead of
the_content(), the first 120 words of the post will be show automatically or whatever you have entered in your explicit excerpt text area on the Write Post panel.
Both methods will show the excerpt on the front page and when you click the title, it will show the full post on the single post page, whether or not you are using the
index.php to generate both page views. The view of the page is based upon The Loop not the template tag you use within it.
If you are trying to generate a specific look for the single post view, then you can use a template called
single.php as discussed in Stepping Into Templates.
Podz mentioned using the
page.php template which is used for generating Pages in WordPress not single post web pages.