I’m not Kayeby, but (in Australia at least) a powerboard is a electrical adaptor that lets you plus four or more electrical devices into one household power socket.
And shower curtain rings.
Shelf liner / contact paper can be useful when moving into slightly-dodgy-but-perfectly-adequate apartments. A box of baking soda for the fridge, too.
If you can find a free evening before you move, and you are moving into a neighborhood you haven’t lived in before, try and find the grocerystore, the gas station, the bus lines, maybe a good restaurant…
It can make that first could of weeks easier if you know where to get milk and bread.
This is one of those “problems” that answers itself in the first day or two.
Whatever you already own can carry you that long. (Just pretend you are in a hotel and you can do without anything.) After that a trip to the store does it.
Don’t move in on the weekend. If you move in on a weekday, it will be much easier to buy what you’ve forgotten.
Oh, oh, oh, the most important one, for which I’ve cursed myself every time I’ve moved:
Somebody at some point in the move or the subsequent building-and-doing-things will slice a finger open. Have a few first aid things handy. It’s not amusing to be bleeding all over a rented carpet floor while somebody rummages through boxes for a few tissues…