Definitely try ToDoList from AbstractSpoon Software. I think it does everything you mention and more. It's Windows only, but it's free, EPL licensed, small, fast, reliable, flexible and will update automatically. You could use it in a very simple way by ignoring or disabling the features you don't need, or you could use the plethora of features that are similar to those you'd find in a full-on project management tool. Over many years I've tried many of the products listed above and many more too. This is the only one I still use.
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Definitely try ToDoList from AbstractSpoon Software. I think it does everything you mention and more. It's Windows only, but it's free, EPL licensed, small, fast, reliable, flexible and will update automatically. You could use it in a very simple way by ignoring or disabling the features you don't need, or you could use the plethora of features that are similar to those you'd find in a full-on project management tool. Over many years I've tried many of the products listed above and many more too. This is the only one I still use.