Hi there - category pages, like the one you linked to, usually do just provide an excerpt of the post.
Most folks who have a blog usually create a page where their blog "lives" - and that page would show your full posts, one underneath the other, with no "Continue reading" links.
If you'd like to set that up, just create a page called Blog (or something else). Leave the page content blank, as this is just a placeholder.
Then go into Settings > Reading - and next to "Front page displays" > "Posts page" - choose the Blog page you just created from the drop-down list.
You should now have a new menu item called Blog, and your blog posts will now be displayed there in their full form.
Just let me know if you need more help.