How can I create a morning routine with my HomePod that would read me today’s weather?
You can surely do it with the Shortcuts app. It's available on iOS and every shortcut you create will be synchronised with your other Apple products on the same iCloud account.

It works well with HomeKit, the default weather app and the Apple Music radio stations. You can trigger a shortcut with Siri, and you should check out the user guide for more infos.