Sounds like you are using the default theme. Just make a copy of that theme first so the next update won’t wipe out your changes.
If you are using the default theme do this:
1. Make a copy of the wp-content/themes/default folder. Call it wp-content/themes/blog-name.
2. Edit the wp-content/themes/blog-name/style.css and modify the second line to read
Theme Name: WordPress Default My Copy
3. Activate the new copy via https://your-blog-url-here/wp-admin/themes.php
4. Now edit the file wp-content/themes/blog-name/index.php and modify line 20 from
<div class="entry">
<?php the_content('Read the rest of this entry »'); ?>
</div>
to
<div class="entry">
<?php the_content('Read more whatever you want it to say'); ?>
</div>
And then you’re done. Naturally, step 0 is back up your files first…