How do I export my Google Fit data to iPhone and keep it in sync in the future?
To answer the first part of your question - the Android part:

You export your Google Fit data by going to Google Takeout.  It is not exported from within the Fit app itself.

Takeout is part of Google's Data Liberation project.  It enables you to download a copy of your data held by Google and attached to your Google account.

The Fit data can be downloaded in TCX or CSV format.