What is the best way to pass values from one form to another with fapi?
You can do a multistep form, there is a clear usage example in the Examples module. Here is an excerpt about it from the module's project page:

The Examples for Developers project aims to provide high-quality, well-documented API examples for a broad range of Drupal core functionality.
The Examples project contains many modules which hopefully illustrate best practices for implementing various Drupal APIs. These modules can be enabled individually, and will typically add menu items to your site, which should guide you through their features. You can then look through the code to see what they are doing.
Developers can learn how to use a particular API quickly by experimenting with the examples, and adapt them for their own use.
Note also that these modules demonstrate internals for Drupal development in PHP (and some JavaScript). Not all Drupal users will need these examples. There are many contributed modules which will do the majority of what a Drupal site-builder could need. Be sure and explore other contributed modules to see if you can find one that meets your needs.