How do I set up Reminders app weekly on iPhone to be shared with family?

Sure. Go to your Calendar app, select an upcoming day for this, and tap-and-hold on an hour slot to create an Event (or Reminder). While filling in the details, you can switch the type for this new addition between Event and Reminder, at the top.

• Events can be “All Day” events, or at a specific time with a duration. Events are intended for where you have to be somewhere. Events can have one or two alerts (notifications).
• Reminders have the start time optional. Reminders stay until checked off; until you confirm that it is done. Reminders are intended for list things that you have to do.

The Repeat option allows you to set a frequency for more of the same with a time interval. Events or Reminders can have a Custom setting for Repeat, e.g. Every 5 days.

Reminders is also a separate app.

Create and edit events in Calendar on iPhone - Apple Support

Use Reminders on your iPhone - Apple Support

… that can be shared with my family.

The Calendar app can have multiple calendar sets, including a named shared calendar.

Share iCloud calendars on iPhone - Apple Support

The Reminders app can have multiple lists, including ones that will be shared, if you choose to do so.

Share lists and collaborate in Reminders on iPhone - Apple Support