Now that I am using the iPhone or iPad with FMSP, I think I need my database hosted on the internet so I can access it from my desktop or device when I am mobile.

This is a situation where you need FileMaker Server. FileMaker Server is designed to host FileMaker files for access across the internet, and also provides automatic backups. You can install your own copy of FileMaker Server on your own hardware, or you can pay an outside company like RCC to provide the hosting services for you. We provide hosting with fees starting about $50 a month, with the database served from Amazon’s high-speed Cloud network.