If you are lucky, you can skip watering your plants for a few days; place the pots in a cardboard box and put the box in your car and go. It is not always that easy.
Some plants get very large while you are living in your home. Some are delicate. And some will have to travel a long way to their new destination.
If you are just boxing the plants and moving them in the car be aware that the light in the car is far brighter than in your home. Most of your house plants will need to be placed on the floor of vehicle or covered to prevent burns. The reason for not watering them first is just to keep your car from getting soaked.
If the plant will moved by the movers you need to do a little more work. First water it and let it drain really well. Then take a plastic bag, what ever size works best for your plant, and place the plant and pot in the bag. Then pull the bag up around the top of the pot and the base of the plant, allowing the plant to stick out. Use some string to tie the bag closed. This will keep the dirt in the pot and the plant in the pot while it is being relocated.
If the plant will be boxed use the trick the orchid companies use. Take a bamboo stake like you use to tie up your tomato plants. They can be found cheap at any garden center. Cut it to about an inch shorter than the inside of the box. Press the stick into the soil near the center of the plant and all the way to the bottom of the pot. Then place your plastic bag around the pot and tie it up over the soil. The bag will keep the dirt and the plant in the pot. The stick will keep the plant from bouncing up and down in the box and getting damaged. Despite the best efforts of the post office, UPS and FedEx not a single orchid in spike shipped to me this way has been damaged and I’ve received well over 100 of them.
All of this works for larger plants as well. If a box is not available put several stakes, as heavy as you need to protect the plant from breaking around the outer edges of the pot. Use cellophane wrap instead of a bag to keep the soil in place. You can wind in through and about the stakes. Then just cut the cellophane off when you reach the new home.