How do I Transfer all Data From one WP Site to Another?

blogzandstuff said:

One simple plugin will do all this for you, called duplicator - easy to use and will move everything for you in a few clicks. no messing with databases - no tech now how needed - if you can read then it can be done and it's free! i use it nearly everyday to build and move sites from one host to another

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I second this. Blogzandstuff brought the plugin up on one of my guides as a recommended plugin, and I managed to easily backup a client's site that was hosted on GoDaddy (their backup system is heinous).

To move your files from one host to another you can use your favourite FTP client.

backlinkingseo said:

To move your files from one host to another you can use your favourite FTP client.

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Incomplete answer: the way you state you won't have your wp-settings, plugin-settings, posts, comments and other database related stuff. As stated several times above, I second the Duplicator plugin.
If you only want some posts, copy the whole shebang to your new site and delete the posts you don't want. If you have access to phpmyadmin, you can always search in the database for posts and remove them in bulk. Easypeasy!