I moved 15 months ago and bought my boxes from Kite Packaging. All the stuff arrived within a couple of days. It was worth the expense because I had a lot of computer equipment and other electronics to shift, and I was moving into rented as I wanted the sale proceeds before I started looking for the new place in earnest. As much as possible went into store.
I also bought a roll of bubble wrap and a huge bag of packing chips, the starch ones not the polystyrene ones. Don't get them wet until after you've unpacked at the other end. If necessary, you can wash them down the sink, no problems, but they also can go in the compost bin if there is such a thing at your new property. I was surprised when I advertised still good packing stuff on Freegle after I'd moved in, and didn't get a response.
The advice on the size and number of boxes to get is good, I had to order extra, and I had too many big ones and not enough smaller ones.
Number the boxes and make a list of what's in which box. Pack stuff according to which room it's going into as far as you can, and label the box and each door in the new house, with the name of the room. For stuff with vulnerable corners – furniture, mirrors, pictures, etc – you can buy corner protectors, mine stayed where they were put, as claimed, I was really pleased with those. If you're taking curtains, leave hooks in and fold the curtains with the hooks inside each curtain.
Pack the drawers of bedroom furniture with whatever usually lives in it plus use up the rest of the space for clothes. Tie twine around each piece of furniture so that it secures each drawer handle and stops drawers coming out in the unlikely event of an accident. This may not work for you, it depends on having furniture with drawers and handles you can tie string around.
Kite's double walled boxes are very good, stronger and more rigid than the thinner ones, better for the fragile stuff, and you'll need a lot of packing tape as well. The fibreglass tape is much tougher than the other kind, but tearing it off a box ruins the box if you're thinking about keeping them for storage. Use a Stanley knife. I bought a roll of tape printed 'Fragile' for the boxes of the electronics.
Take tea, coffee, milk, sugar, kettle and enough mugs for you and the removal men, in your own car. They'll appreciate being offered a cuppa even if they want to get away quickly. Take a torch with you, and have any tools for assembling flatpacks easily available. Put screws for furniture, clothes rails, etc in small plastic bags and tape them well to whatever they're used in.
Do you know which supermarkets deliver to your new address? Put in a grocery order to be delivered the afternoon or evening you move in, but go out to a decent pub for a meal on the day, you'll need it. 
Get quotes for moving as soon as you can, you'll have to have a fair idea of the volume you'll be moving, and book your movers as soon as possible.
Google for house moving and packing tips! For telling utilities and others you're moving, there's a website called Am Moving which I used. I did wonder if I would get a load of spam as a result of it, but that didn't happen. It was helpful. They'll talk to you about changing your electricity/gas suppliers and quote you figures for different companies – they get commission on that. I switched suppliers and am paying a lower tariff than I was previously.
It's making me tired all over again just thinking about it! And it's nearly three in the morning, too hot to sleep.