confused on theme files

I am building a theme and it is going good. my issue now is this. it’s not really a blog website but I am going to add blogging support. I created a page called home.php, got all my content into that. I wanted to make a blog page so I created a category called blog, then made a test post and assigned it to the blog category, then added the blog category to the menu. When I checked out the page, I found that the page that is displaying the post is the category.php. I thought maybe the index file would display it. So when does the single.php and the index.php kick in? How do I assign a blog page? if someone just clicks on new post without using a category, how do I get that post to display?