The example you like has a home (static) page and a blog page. You can keep the home page you have but it will have to be a static page. You can't have 2 blog pages per se.
You could set up a category page but that would only show certain categories while the blog page shows all. That way you could have multiple blog pages but you would have to change categories when you wanted a recent post replaced with another. I don't think that is the best way. http://codex.wordpress.org/Posts_Categories_Screen
So your static page will have everything you have now and just show parts of the most recent 3 posts you have made which is very much like a teaser. Part of the post - a teaser so they read it and click on read more, which takes them to the single post page like it does now.