Given the luxury of space, the easiest way to relocate aquatic pets is to prepare the new tank and move livestock and a mature filter simultaneously. For many of us upgrading a tank occupying the same place, a temporary third container may be needed. This needn't be a glass tank though, and any non-toxic watertight container that can safely be used together with heating and filtration can be used.
With freshwater fish, filling the new aquarium with dechlorinated water (as used in the existing tank) adding a heater and means of water movement should make it safe for use with fish twenty-four hours later when the mature filter and the existing livestock can be added. If possible, transferring water from the old tank will often result in an easier transition for your pets. Keep feeding light for a few days whilst filter bacteria recover and consider adding a product such as Aquarium Munster's Bactosprint to ease the transition.
As most marine systems rely on live rock (or a ceramic alternative) this can be moved in the same way as a mature filter. For territorial fishes such as marines and cichlids, duplicating existing layouts can help preserve pair bonds and hierarchies.