This is a great question and you will probably get many good answers. I recently set up a new 55 gallon aquarium from scratch and I added my fish the same day. The key factor to keep in mind is that you need to have a good amount of nitrifying bacteria present in your tank. You can sort of naturally cycle a new tank or new substrate two different ways.
The first way is the way I do it, I go ahead and remove as much water as I can store out of the desired aquarium, I keep the siphon high in the tank to try and remove the cleanest water. I personally have a 15 gallon bucket and two 5 gallon buckets that I use when I do something like this. After draining a good portion of the heater your going to want to go ahead and get all your fish,snails,shrimp,etc out of the tank and into your biggest bucket, I personally also put in a few fake plants when I put them in the bucket so they can hide and feel safe, you can put a small powerhead or an air stone if you want in there as well. Then you will want to drain the rest of the water out of the tank and scoop out as much of the substrate as possible, being carful not to scratch the tank. Once you get most of it out I would suggest taking the tank outside and washing it out with a water hose, this will remove all the remaining substrate and will not scratch the tank in the process. This is also a good time to give the interior of the tank a good scrub down. After that, rinse your new substrate, unless it is instant cycle or live substrate, and fill your tank with the desired amount. Then I would add some treated water to the tank and start running my unwashed filter system from the previous set-up, run it about an hour. Then I would add as much of the reserve water from the previous tank as possible, at this point I like to just add some Prime and stress coat, you can also test your water at this point. Now it should be safe to add all your critters.
The second natural way is easy but just take some extra time. If you have the extra time all you have to do is clean your new substrate, unless it is live or instant cycling, then put it into a five gallon bucket, I layer in a few air stones as I pour in the substrate, then add about three large handfuls of your current substrate onto of the new stuff and cover with treated water, once it's covered with water you can pump air to the air stones on a low setting. Your new substrate will become "live" in as little as five days.
If you want to make sure you cover your bases I would suggest going to your LFS and picking up a bottle of nitrifying bacteria, they are commonly labeled as instant cycle or something like that. This will ensure you get a large amount of good bacteria in the tank. I hope this helps, keep us posted on what you decide to do.