On the old server loginto your WP admin panel then go to tools, export. Export what you need once that is saved to your desktop you can go to tools, import click on the one that says WordPress Import posts, pages, comments, custom fields, categories, and tags from a WordPress export file.
Not sure what it will import but you can try it.
Another thing is you can do a mysqp dump and then import it into the new server or use a graphical program like MySQL Administrator. You can find it on the MySQL site.