I want to add an automatic date stamp when an opportunity moves from Stage 1 to any other Stage that isn't closed.
You can use a Process Builder/Workflow Rule/Trigger to achieve this. Just create a custom Date/Time field, choose the appropriate method you want to use (e.g. Process Builder), define the action to fire when it meets your criteria (e.g.
ISCHANGED(StageName) && NOT(IsClosed)), and use an action to set the Date/Time field.