I would like to customize the contact record page to display different fields under different tabs on the page for organizational purposes, instead of displaying all of the custom fields under the standard 'details' component. I understand ( i think) how to divide the page into tabs (see photo below), but how do I go about putting different contact fields within the different tabs on the record page? For example, I want different fields under the 'home/correspondence' tab than those that display under the standard 'details' tab or that I would put in any other tabs I have on the record page.
There is no declarative functionality available to put fields on different sub tabs of a Lightning Record Page. You would have to create either a Visualforce page (enabled for Salesforce 1 and Lightning Pages), or a Lightning Component, with the fields that you want to show on the tab. Then remove those fields from the Detail section via the Page Layout editor. But the only out-of-the box way to put editable, record-level fields on a Lightning Record Page is with the Details component.