Re: bbpress to bbpress
First, I think I would make sure the bbPress version are the same. If they’re not, first backup your MySQL database then upgrade bbPress at your current host. That way you have the same version of the code and the database.
Now, dump the database, paying attention to the MySQL version in case you need to specify any compatibility options for the new host. If the version are close to the same, you’ll be OK just dumping it.
Now that you have the bbPress code at the new server and a dump of the database, just import the old database on the new host, and you should be done.
If the domain name changed, you have a couple more things to worry about. But you didn’t mention that: just a new host.