Do you have a custom or hacked theme? if not then do the following:
From old host:
The a backup of all posts etc from:
WP Admin => Manage => Export (under options make sure that all author's is selected)
Download Export File
For new Host:
Upload Wordpress via FTP/Or do it from the CPanel or whatever your host has provided (if there is an option for a one-click/application install of wordpress.
Upload Theme
Delete the preinstalled Post and Page.
Go to:
WP Admin => Manage => Import => Wordpress => Select File on Your PC that you downloaded earlier and hit Upload and Import, its pretty self explainatory from thereon.
If your host is doing it for you, you can always take that option, though I would recommend trying it out on your own, if it works, that would be knowledge n that cant hurt!
Cheers!!