Water won't help cycle the aquarium as the bacteria we need to grow live in the biofilm which is firmly attached to surfaces. They do not live floating in the water.
The only time using old water is useful is if water changes have not been done regularly and the water parameters in the tank are significantly different from tap water.
Do you intend keeping the original tank running, or moving everything to the new tank and closing the old one?
If you don't intend keeping the old one running, you just need to move everything over to the new tank. The current filter should hold the majority of the bacteria in the tank, though there will be some on other surfaces. If you intend using a different filter in the new tank, move all the media from the old filter into the new and fill any spaces with media which is supposed to be in the new one. Sponges can be cut up to make them fit; ceramic media can be put in a mesh bag.
But if you intend keeping the old tank running, the easiest thing to do is take up to a third of of the media from the old filter, filling the spaces with media from the new filter, and do a fishless cycle. Because the old media holds a lot of bacteria, just not enough, the cycle will proceed very quickly.