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For the past 6 months or so, my wife and I have been using an app called "OurGroceries" to manage our shopping lists, and I recommend it highly. Through a smartphone app, we can easily add things to various shopping / errands lists, see instantaneous updates as we shop, and a lot more. The app is very easy to use, both in adding and crossing off items on your shopping list - we use the iOS version on our iPhones, and there is an Android version, as well.
In spite of the "groceries" reference in the title, you can use this for just about any shopping or errands list. As you can see in the screenshot, we've set up lists for a number of our commonly-visited stores, as well as a general purpose "Groceries" list. As we think of something, it is very easy to add it to the list, and it automatically shows up for the other person.
As you use the list over time, you tend to repeat items, and the app makes it easy to add those items to the list. You can either start typing and it will autocomplete based on your history, or you can tap on one of the crossed off items (as seen below) and that item will be moved back to your list. I also find I am less likely to forget things if I scan through old items to jog my memory.
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Another cool feature is the ability to set up recipes (we only have one, at the moment as you can see in the first screenshot). With recipes, you can bulk-add items for a specific recipe or project. Once you define the recipe, a single tap will add all of its items to the appropriate lists. The recipes don't have to be literally recipes - for example, you might have a "Backyard BBQ" recipe that includes a bunch of items you want to stock up on to prepare for an upcoming event.
OurGroceries has two versions - a free, ad-supported version and a paid, ad-free version (paid via an in-app purchase). After seeing the value a few weeks in, I paid the $4.99 for the ad-free version and I haven't regretted it.
You can also edit and manage your lists via OurGroceries' website. We don't really use it that way, but it is good to know that the option exists.
If you need to coordinate shopping and errands with other people, give the free version of OurGroceries a try - I think you'll like it.