Can something repeat every week or every month?

Yes. Open a checklist item that has a due date and choose Repeat. The options are:

• Every day
• Every weekday, Monday to Friday
• Every week, on the same weekday
• Every month, on the same weekday — “the fourth Saturday”
• Every month, on the last such weekday
• Every year, on the same date

The pattern is read from the date you set, so you never have to describe it twice. Ticking off a repeating item schedules the next one.