Key to solution dependent upon version of the old Moodle. If 1.9.x users will not come across even with a full (including users) backup of 1.9 courses. So what is version of old Moodle server?
Suggestions: On the fresh (new install) setup imap authen that matches what you had with the old server. Test login with a known imap account. If that works, then ... take one course you desire to move to new server and make a full backup (including users). Restore to new server. This assumes the old moodle courses are version 2.
If you find a way to import just users, that sets them up for logging on, but doesn't automagically re-enroll them in courses to which they were enrolled (I think).
Yes there is a lot of clean up that needs to be done on a 1.9.x -> 2.x migration but that's on the admin side of things. All in all ... considering factors where you'd like to end up, removing courses might be the least disruptive to users. One could, in the new migrated system, create a category called 'remove' ... then move all courses to be deleted in the 'remove' category. Then delete the category. Moodle 2.6 will prompt with options to move to another category area or delete them (cannot be un-done).
** make a full site backup (code directory, data directory, dump of DB) before attempting above.
This could be some heavy processing server wise, so might not be a bad idea to increase PHP settings for executing a script, max memory a script can consume, etc.. I can't tell you what values ... and some might tell you that is not necessary, but think that's what I'd do anyway (best to have more than enough, than not!).
'spirit of sharing', Ken