Krazyvan answered your question almost exactly as I would have. Thanks for saving me some time!
You can try to lift the old pond out but I expect it may become stuck. Also, there will be the most gunk on the bottom which is not great for the animals to sit in. So, I suggest removing and netting out the larger animals or at least those you can easily get. Then, you can pull the pond out with the more stubborn animals down in the bottom gunk. I suggest using kiddie pools or plastic storage tubs to save water from the old pond to refill it when you set it aside. If you stir up too much stuff in the old pond, and it's too dirty, you'll want to clean it all the way out. See
http://www.fishpondinfo.com/myfish/clean.htm for how I clean out my 153 gallon pond. In your case, you're cleaning your old pond and setting it back up while the new pond is being made.
Once the new pond is ready, fill it up and run the new pump and filter for a day and aerate the new pond well. Be sure to add dechlorinator if you have city water and maybe a little pond salt (0.05% at most). Then, the animals and plants can go right into the new pond which can't cycle until they are in there. Put the old filter's bioballs into a bag in the new filter so it's innoculated with the good bacteria.
If you're in a rush, you can do the above in a day but aerate the new pond water heavily for at least two hours before putting in fish. See
http://www.fishpondinfo.com/fishcare/water.htm#gas for the reason why you need to aerate.
Adding water from the old pond to the new pond won't help it cycle much faster as the nitrifying bacteria are attached to surfaces (bioballs, side of the pond, plants, etc.) and not so much in the water itself. Adding the old water will introduce the microlife that was in your old pond such as microorganisms, algae, and any bad stuff too. I wouldn't bother putting in the old water.
Good luck!