There is a one-click migrate for DreamPress – at least it is advertised as such. You can start with that. I don’t use DreamPress so I can not state an opinion of the DreamPress product.
Wordpress’ life exists in the database. If you have access to the database backup then transferring the Wordpress site is relatively easy. If you don’t have access to the databasse backup then you can use the built-in database dump. However, it just dumps the WordPress items, not the tables for the various Plugins.
There is a plugin that seems to work well. It is called WP-DB-Backup that you can install with nothing needed but admin level access to Wordpress. You don’t need admin level on the account, nor a shell account, nor control panel access.
This plugin will generate a dump of the database tables for not just WordPress but also all of the plugins in a form that can be reloaded by MySql (or phpMyAdmin).
You still need to copy the wp-content uploads and gallery data, but that can be done with FTP.