First, upgrade to v1.3.7.1 or newer (since that's required for PHP 5 support). As of today v1.3.8a is most recent.
https://www.zen-cart.com/tutorials/index.php?article=98
Then apply all the relevant security patches for the version you've upgraded to.
http://www.zen-cart.com/forum/showthread.php?t=131115
Then TEST TEST TEST to make sure it's all working as desired.
Then move to the new server: https://www.zen-cart.com/tutorials/index.php?article=100
DO NOT MOVE to the new server before upgrading ... otherwise you'll have numerous error messages and checkout won't work, due to PHP5 incompatibilities with the older version you're using.
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