Cycling your tank is just the process of establishing colonies of bacteria that will convert ammonia => nitrite => nitrate.
The greatest concentration of this bacteria would be in your filter. So actually if you want to setup a new tank, you want to at least initially either take the old filter with you or at least try to some how reuse the media in the old filter to seed a new filter.
Tank decorations will also have some of these bacteria on them, but not in as much concentration as the filter.
So once your tank has cycled, anything that you remove from the tank is going to take some of your bacteria colony away with it.
If you attempt to setup a new tank, the more things you can take with you from your older tank will make cycling the new tank that much faster.
Perhaps the best way of setting up a new tank would be to set up the tank with the filter you eventually want to have in it, and then also run the old filter in it as well. If you did a fishless cycle, simply feed the new tank ammonia to allow the existing colony from the old filter to migrate to the new filter. If you already have fish, don't add any more, and again run both filters in the new tank. Within about two weeks, the new filter should have enough of a colony built up that you can remove the old filter.
What is even better (if it doesn't take up much space) is to keep both old and new filter running in the new tank. That way, if you ever get a sick fish, or when you get a new fish, and you want to quarantine that fish from the others for a few weeks, just pull out the old tank, fill it with conditioned water, and move the old filter back to the QT tank... instant cycle.