I don't need speed nor is this a "relocation" type of shipping.
It is a lot of work for all parties involved and that is why even small details matter. In some cases it is cheaper to ship using a complete container, even if you have to fill it with packing material to stop things moving around.
My family have moved quite a bit of furniture internationally and considering the costs, insurance, damage, claims, etc. Unless you are talking about very high value antiques it is not worth the time, effort and cost.
In my experience people want new, modern looking stuff. A couple of years ago I had to sell beautiful estate furniture which cost tens of thousands to buy. I only got a couple of thousand after a lot of haggling, etc. I didn't even recover the shipping costs that my family had paid decades ago.
There is a minor chance it will get lost overboard or stuck in a canal. So you should arrange specialist insurance.
The reason everyone wants to give you a personal quote is that end users are unreliable. And you don't know how things have to be packed. And you don't know how much free space there is on an upcoming ship. And you don't know how easy your home is to get a shipping container to. And... It goes on.
I'd start by looking at expat forums and seeing if people have decent/recent experience with a company.
But… unless this stuff is either expensive, irreplaceable, or sentimental - you're probably better off selling it locally and buying replacements.
In order to give you a quote they will need the weight and dimensions of what you are shipping, the pickup and delivery address, a packing list, and a commercial invoice stating what each item is and what it's worth. You've also got to think about how you will load everything. Your options are lifting by hand, using a forklift, or you'll need a truck that has a tail-lift. If you've got all this information beforehand it will make things easier.
If you're putting thing's in a crate like another poster suggests then you'll have to make sure the crate is heat-treated and has the appropriate stamp on it.
We just shipped something at work from the UK to the West Coast of the USA (sea-freight) and it cost about US$1700. It was a crate of about 1 cubic meter and weight about 350kg. This was door-to-port not door-to-door though.