I suggesting using a tool like VaultPress, pay them their $15/month and follow their backup/restore process to move the site. I've done it successfully myself. It ensures all your store pages and categories, etc. move along with the main site.
You should really have a solid WP site backup anyway. Server backups, especially from hosts are not very reliable.
If you want to do it by hand you will need to copy over the entire *_store_locator table AND make sure you get all the custom post types moved over with the SAME EXACT PAGE ID... or you will need to re-create all the Store Pages after you move. Personally I'd delete all the Store Pages BEFORE moving the data and just re-create them on the new site. Should be a 1-click operation after the move.
All support for SLP is moving to the CSA website. Posts on that site now take precedence and get my attention first. Visit the new support forum here: