Unlike esmtp, using your ISP's SMTP server with postfix will still allow local mail to be delivered. Any hosts listed in the mydestination option will be exempt from the SMTP relay.
Set up ISP info
Create /etc/postfix/sasl_passwd, like so:
mail.my-isp.org username:password
Protect this file, since it has a plaintext password:
chown root.root /etc/postfix/sasl_passwd && chmod 600 /etc/postfix/sasl_passwd
Create sasl_passwd DB file
postmap hash:/etc/postfix/sasl_passwd