At three weeks to go, you'll hopefully have finished packing your cupboards - you'll be able to start packing any non essential belongings and put them away too. If you're painting any rooms, its a good idea to try to do so in the next few days, so its all ready, and order any packing supplies you want or need. Ordering as far in advance is important because you'll have the supplies on hand.
You can start investigating schools in your new area and list any questions you may want to ask. While considering which the best school in the area is, you also have to consider how accessible it is, whether there's a school or public bus route to take your children there and whether you need to pay for this.
Your quotations should be coming in now too, from those that you've requested them from, and they should be making appointments to evaluate your belongings, to decide what size of van you'll need and how many men you'll need to help you move - or just the size of van you'll need. You should have chosen your new utility provider by now, and have any forms you need to fill out for your bank, so that there's no delays in changing your details to your new home. If you're letting, you should also have a lease by now, with details of what you are going to have to take care of on moving in.
Your Internet is also something you should start closing down, and let them know when you're moving – if they have to come pick up (or move) your equipment, now is a good time to arrange this. Internet service providers have a minimum notice clause, so its good to make sure that you meet this, or you may be over billed, or fined.