first things firt. Are you running the same version of SeoToaster on the old and new server?
I moved a SeoToaster webiste from two different VPS running cPanel and Apahce, but under the same hosting, so same structure.
You should check if the path to core is the same under new and old hosting.
for both sever
if so, you can basically copy-paste everything, and just change the domain pointing (like if you have a DNS redirect, or change the folder it points to, in your sever setting).
With copy-paste I mean:
// old server
1. copy old DB (mysql export from phpmyadmin)
2. copy old files (make a .tar.gz or .zip package)
// new server
3. create new db with SAME db_name, db_user, db_password as the old one
4. import in the new db the old one (mysql import from phpmyadmin)
5.create new folder, with the same name as the old one
6. upload old files and uncompress
Well, to keep the answer short, that should be a starting point. Good luck, keep in touch and update the Question!