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...

https://support.office.com/en-us/article/Use-the-List-web-part-on-a-modern-page-ef0a1b80-f8b3-443d-b04a-1e76c70b5537

https://support.office.com/en-us/article/Use-the-Document-Library-web-part-on-a-modern-page-a9dfecc3-2050-4528-9f00-2c5afc5731b0

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