Moving to another host; which tables to move in a database

I never personally tried this, but your images and posts reside in your {prefix}posts (wp_posts) table. I would also copy {prefix}postmeta (wp_postmeta) over.

So copying over those table should do what you want.

Keep in mind that this won’t copy your categories and tags (meta), and your comments (comments) they reside in other tables, you could find those easily they have meta and comments in their names. Copy those too if you want that.