First thing is to organise a full day of childcare for the children away from the flat. They will get in the way and slow you down completely. Also, it can be upsetting for them to see toys etc., packed away. Plan on trying to get it all done in a day, or have childcare arranged for a second day too. Best to have them out of the way for the move-in too, because you won't be able to keep track of them with the door open and movers going back and forth. Moving can be very dangerous as DCs can get lost, play with stuff normally out of their reach like cleaning supplies, or can actually trip up movers carrying large pieces of furniture. If you have a pet, same advice goes. Pets can easily be a nuisance and have been known to get lost.
First thing is to get supplies: cardboard boxes and plastic boxes, packing tape and markers, newspaper, bubble wrap, scissors. Stock up on bin bags for things you'll be dumping.
Gradually empty your fridge and pantry, beginning soon. Buy paper plates and cups for when the good ones are packed.
Set aside kitchen and personal hygiene essentials, pantry items like teabags, etc. and a change of clothes, nappies for everyone. These won't be going in the van. Set aside cleaning items for when you've just moved in to make the new place habitable.
Have food for the day set aside, and something for the movers too. Bottled drinks are great.
Then gather your important documents and put them in a plastic box, clearly marked. These don't go in the van either. Same goes for your DCs' special toys.
Go through each room with your boxes. Put heavy things in small boxes (books, cds).
Put clothes in plastic boxes, same with bedding. Put the children's special toys in their own plastic box (favourite stuffed animals, etc.). Put the rest in big boxes. Mark mattresses with the beds they belong with, maybe a number system would work.
For items like the tv or whatever larger home electronic things you have, it's nice to have the original box the items came in but if not, use A LOT of bubble wrap for these things, same for kitchen appliances like coffee maker, slow cooker, etc. Too much bubble wrap is far better than not enough.
Use medium and small boxes for china, which is heavy, pots and pans, baking ware. Use large boxes for things like table lamps, with padding in the form of children's soft toys or something similar. Label everything with the room it comes from and what it is, and where it's going, 'fragile' as needed, and an arrow indicating which end is up.
I love the idea of having a number for each room and a corresponding number in the new house, but this mightn't be possible until you know where you're going.
In general, heavy things go in small boxes, with lots of tape, and the bigger and lighter individual items go in bigger boxes. I would put clothes and bedding in plastic boxes. The interior of vans can be damp depending on the weather. The boxes may spend a bit of time standing around outside before being carried into the house or into the van.
Try to keep the boxes out of the way of the exit route for heavy furniture. Corners of rooms are good. Try to keep stacks of boxes low to prevent the DCs from tipping things over on themselves.
If you have carpet, it may end up looking really horrible after the movers have walked in and out all day so you might like to check what your landlord will accept in the way of normal wear and tear for the carpeting. Most, I suspect, would be cleaning carpets before the next tenant moves in anyway, but you never know what they might do with your security deposit.