Sharepoint - Adding web part to page in Modern view

A modern page does not allow you to add a listview webpart to it. If you need this functionality, a custom SharePoint Framework webpart would be the answer.

Also, it is not possible to switch a modern page to a classic page. If you need the classic page functionalities, webpart or wiki pages should be added (either in SitePages or in Pages library).

Update It appears MS have released List/Library webparts for modern pages just recently. Rollout will start for first-release tenants, so it might be a while to see it available everywhere though...

Customization in modern UI is not possible yet. You can create webpart or wiki pages using SharePoint Designer and then add webparts/views.