I recently shipped some household items to Vancouver from the UK using Pickfords and was very happy with them.
I used a company in Birmingham (cardboardboxes.co.uk ?)for custom-sized boxes for a couple of furniture pieces - I had a few issues with time delays with them (I was trying to arrange from Canada) but ultimately they delivered the flat packed boxes for a reasonable cost and the product was great. Ordinary cardboard boxes won’t cut it - you will need double or triple walled boxes. I packed myself but for a whole house I would get the moving company to do much of it.
As PP have said , don’t ship anything Ikea or similar. I wouldn’t do beds or sofas either. I do really regret not shipping a lovely dining table and sideboard- I decided it wouldn’t be worth the cost as I could buy something new for same price as shipping. Turns out I couldn’t find anything new that was the same quality (and had the same memories) as that old table .
IME, the things that make a home are soft furnishings (just take the cushion covers), books, art, sentimental items, and everyday kitchen items like favourite mugs, crockery, bowls etc. that are just “home”, and the odd piece of furniture like a favourite chair.
Don’t take any electronics (transformers are a pain and ugly). For sports equipment like bikes, unless it is highly specialized, then don’t bother. Be ruthless!
That said, it is very overwhelming to arrive and start from scratch furnishing a home (especially when you have kids, new jobs, etc, etc) Maybe speak to the shipping company, set a budget / size of container then fill with as much as you can. Remember, unless you are buying from Ikea, new furniture needs to be ordered and can take months for production and delivery.
I recently moved house and had to buy a lot of new furniture- I highly recommend Article for decent quality and good customer service. Very disappointed with CB2 for quality- overpriced for what it is and customer service not great. Crate and Barrel - in store customer service great. EQ3 also decent quality and good customer service. Simons (Montreal-based department store chain) also good for house stuff. Avoid Hudson’s Bay - terrible customer service.
As PP said, Canadian customs do require you to list every single item in every single box. Do this as you pack, not by memory later as I did 😁. I can’t recall exactly but there is some issue with bringing mattresses in to Canada. The shipping company will be able to give you an info package on that stuff in advance or DM me and I can dig out the info they sent me. It took about 6-8 weeks for my stuff to ship from UK to my doorstep in Canada. I did have to go into the customs office in Vancouver to get a form stamped before it was released from customs and delivered to my home but this was a quick and easy process.
Good luck with the move - feel free to DM me with any questions 🍁