First backup everything on your client's server. Backup the Showkase installation on your own server – you won't be changing this but good to have a backup.
It's usually possible to copy the old Showkase installation to the new server, run Showkase setup again and republish. You can try this. However if something goes wrong then it may be hard to understand the problem so here is a safer step-by-step method.
Install Showkase on the new server and check it is all working.
Each of your web pages has its own folder with the name you gave it – copy one of the page folders from the old installation to the new.
Go to Showkase > System > Repairs and click Repair
The page should appear in the Showkase Pages screen.
Publish and check the page is ok.
If you get error messages about permissions then change the permission on the page folder and contents to 0777.
Move your other pages across in a similar way.
jack