Events with specific times go into my Outlook / Live.com calendar that syncs between all my devices and all my wife’s devices. Her events go in her Outlook calendar which syncs the same.
Recurrent tasks that don’t have fixed appointment times (e.g. clean the HVAC filters every month-ish) go into our respective Outlook tasks which also sync across all our devices.
Both the above fire off reminders as appropriate so stuff isn’t forgotten.
I use OneNote (which also syncs between …) for a vast array of general knowledge & records management. Specific to the OP, I/we maintain two pages called “Shopping List” and “Things to Do”.
Any intended purchase from groceries to clothes to hardware store to … goes on Shopping List, grouped by the likely store category to get them from. Items are deleted as they’re loaded into the shopping basket or put away at home.
If it’s not a purchase, it goes on TTD, with the more urgent stuff on top and the round tuits on the bottom.
Since I always have my phone with me I can add an item of whatever sort to the appropriate app whenever the requirement emerges. And I can always cross them off whenever they get done. And we each always have a complete picture of what’s upcoming, impending, due, and overdue. Naturally a few barnacles settle into the bilge and sit there for months. Those are eventually deleted as irrelevant.