OK - this is a BRIEF version and ONLY applies to UN-MODDED YaBB forums

Download a clean copy of the right version of YaBB from SourceForge

1. BACKUP the current server using the YaBB Backup (make a full backup with ALL options selected). Download that backup.
1A - place old server in Maintenance Mode.

2. Open up the Quick Guide in the Download Package and you will find it contains most of the information you need to get YaBB installed including all the CHMODs,

3. On the NEW sever, do a clean new install of the correct YaBB version. MAKE SURE IT WORKS, post something, create a new user, and get the mail settings right. (they might be different) and make a note of those settings.  Reset the Start Date to match the OLD forum.  If you can post and work - you are ready to switch. Place New Forum in Maintenance Mode.

4. Locally extract the backup and locate  the needed folders from the backup.

cgi-bin/yabb2/Boards/* (all files)
cgi-bin/yabb2/Members/* (all files)
cgi-bin/yabb2/Messages/* (all files)
public_html/yabbfiles/Attachments/* (all files)

Read the upgrading from YaBB 2 part of the Quick Guide. Ignore the parts about the settings files, menus and conversions.  You are reading it because moves are the same as upgrades, except (obviously) there's no data conversion.

You are now going to upload the above folders via FTP, and check the File Permissions (CHMOD) so they are all the way YaBB likes them.

Presto Shazaam - you MAY be almost done!  

Login as admin, and use your 'old password' - the admin.vars file will have been over-written

See what happens ---

If it 'all looks good' - hen go to

"Admin Center" => "Maintenance Controls" and run the "Rebuild Notifications Files" first!
Now run the other Maintenance Controls starting with "Rebuild Message Index" going through "Rebuild Members History."

I'm pretty sure that it. (but I am running way low on sleep, so ...)  

Good Luck, and holler back!

Is your new install working OK???

skins? you mean YaBB Templates?

if so, they need to be DL'ed from the current server and uploaded into exactly the same locations on the new server. (of course fixing their permssions after uploading.
There are often two sets of files; the templates themselves found in ./Templates under /yabb2 and the associated Icons and graphics whihc will normally be in a folder with the same name as the matching template under yabbfiles/Templates/Forum. Unless there is also a custom Admin Center, ther won't be any additional folders under yabbfiles/Templates/Admin.

I have to check if there's anything else to be done, and I think there is; a configuration file perhaps.

but, for now - that should getting things started.

BTW, the Codex is working again