Cats and dogs notice the cartons appearing weeks before you move, and they read it as threat. Most of the distress on moving day was set up in the fortnight before it.
Before the move
Bring cartons into the house gradually rather than all at once, and let pets investigate them. Keep feeding, walking and sleeping routines exactly as they are - the routine is the reassurance.
Get the paperwork in order early: a fitness certificate from your vet, up-to-date vaccination records, and for interstate movement a transit permit. Microchip details and identity tags should be updated with the new address before you move, not after.
On moving day
Put your pet in one closed room with their bed, water, litter and a familiar toy, and tape a large sign on the door. Tell the crew about it explicitly. The single biggest risk of the day is an animal slipping out through a door that is propped open for four hours.
If you can, board them for the day or leave them with someone they know. It is kinder than a room full of noise.
Travelling
Short distances: a secure carrier in your own car, never loose, and never in the moving truck. Long distances: a dedicated pet transport with a handler, a ventilated IATA-compliant crate and planned water stops. Sedation is generally a bad idea - it interferes with temperature regulation - but ask your vet about your specific animal.
At the new home
Set up one room first with their bed, bowls, litter and toys, and confine them there while the rest is unloaded. Let them expand into the house at their own pace over a few days rather than giving them the run of it immediately.
Cats particularly should stay indoors for at least two weeks. Their homing instinct will try to take them back to the old house, and cats have walked astonishing distances trying.
Expect a week or two of odd behaviour - hiding, appetite changes, accidents. It usually settles. If it does not, see a vet.