You can’t add posts to pages. Pages are generally for static content, though if you want to add new content to a page, you can edit that page. But your blog can only have one regularly updating blog page.
If you want post to go to a certain page, then you need to assign a category. People clicking on that category in your Categories widget will then be presented with a page with all posts witin that category. That is how blogs are supposed to work. However, your front page will still by default show all posts. As a workaround you can use a static frontpage, as rosclarke mentions, and then the Categories widget. That way people will only see the posts in the different categories.