I have moved many times and I usually have to start with new everything, new substrate, new water, new filter materials because I go from Ohio to Maine ( 2 day trip) and then back to Ohio from Maine.
I have never had a problem with new tank syndrome, but I stay on top of things. When I set back up, I do frequent water changes, over feed fish for 2 weeks until they get settled in. Now that being said...that is an extreme move example to ease your fears of a short move.
I would say for a local move if you can, keep your filter for the media and your substrate as those will carry your beneficial bacteria. Some say take some tank water, but that is just one more heavy smelly thing to carry. Bacteria growth is extremely quick and should replenish your tank in no time, just keep an watchful eye out for awhile.
As for moving the fish... I have used coolers lined with kitchen trash bags that do not carry a fragrance, new 1 gallon trash cans, mason jars, it all depends on your fish. My plecos are housed separately because of their hard fins can damage other fish when bumped..think pot holes...my Malabar Puffer goes in a mason jar because she is so tiny and bumping might harm her...get my line of thinking here?
I hope this helps
