Split blog posts over multiple pages
The core 'book' module will do what you're after.

First, configure the module so that the content type you are using for blog posts is allowed to be in books at admin/content/book/settings.

Second, understand that a book is a bit like a menu, each page is a single node and the book simply provides the links between those nodes in order.  When you create or edit the node intended as the first page of your book go to the 'book outline' settings at the bottom of the node creation form and select create new book.

To add pages, create or edit the nodes that are to be the next page(s) and select the book you just created.

Finally there is a block that the module creates called 'book navigation', move that block to where you wish the 'next' and 'previous' links to go.

Some styling will probably be required if you want it to look like the example you gave.