You can use FTP or your hosts' file manager to move your vB files from one server to another. You need to download all the appropriate vB files and directories (making sure to retain the current directory structure) then upload these files and directories to the new server. Make sure you transfer all the image files (.gif, .jpg, .png, .ico, .attach, .thumb) in Binary format and the non-image files in ASCII format. Most FTP programs are set to recognize the correct file types automatically but it does not hurt to double-check.
If you are storing attachments, avatars and user profile pictures in the file system instead of the database (the default storage method) then you will need to move those files and directories over as well. As with the old server these directories will need to be made world-readable and writable, (chmod -R 777 on *Nix systems.)
Also since the exact path to these directories will very likely be different on the new server, you will need to reset these paths in the Admin CP after the transfer is done and your forums are up and running on the new server. You do this here:
Admin CP -> Attachments -> Attachment Storage Type
Admin CP -> Avatars -> User Picture Storage Type