Hello Laura,
That's a big move you've got ahead of you! But no worries, you'll get there.
Make lists. That way you don't have to remember everything, and you can check off or
score out things that have been done or decided to leave them out.
I'll start with the easier stuff: chicken and ducks can travel in cardboard boxes. Make sure there are enough and large enough ventilation holes in the upper part of the boxes. If you'd be moving in the summer I'd suggest travelling at night, but looks like your moving is going to be in the winter?
Once arrived, you should keep the chicken in the coop for at least 24 hours, 48 is better, to let them set their homing-radar

I don't know about ducks.
You'll have to lure the cats with something they can't resist and perhaps wear welding gloves when catching them (or are they all right with you handling them?). Be prepared to hear loud protesting during captivity... Cats need two weeks of inside time to reset their homing radar. Unless you have the feeling they're settling into the situation earlier... I've lost a cat before to moving house. The cat just wandered off and never came back.
Maybe start looking for a large styrofoam box for the koi to travel in? Fish shops might have them for keeping their merchandise cool. You'll be needing it/them for keeping your koi at ambient temperature. You'd be putting them in a large (or two large enough) plastic bags 2/3 filled with water from their pond and 1/3 with air.
You'll need to think ahead where to put them once arrived and prepare that at least two weeks ahead of time, for the biome to settle before the fish arrive.
This coming autumn, dig up all the perennial veg and flowers you want to take with you, and plant them temporarely in pots or even plastic bags. Keep them fairly tight together in crates, they'll support each other and buffer temperatures. Keep in mind you don't have to move ALL of your plants , just the best individuals of each species you want to save. You can always multiply them later.
Berries, bushes and trees: in their established, mature form they really don't like moving.
Berry bushes; maybe you can dig up and divide the best ones, making many out of one or two bushes?
Nut bushes; can you take cuttings at the end of this summer and root them in pots when the weather is still mellow?
Trees; ...cuttings?
Remember; it's going to be a new start! Leave space for new stuff to come to your life!! This is your Great Opportunity to choose ONLY that what really makes you happy (or any strong reason to carry something to the future) and gift, sell or recycle/discard the rest.
Good luck!