You should set up the larger tank as soon as possible and, yes, let it "repose" for a few days. Beyond that, there are 2 basic things you want to ensure. First, you want the new water to be similar to the old water, so you don't send him into shock. Second, you want to preserve as much as possible of the beneficial bacteria growing in the old tank. These bacteria grow on all the inside surfaces of the tank and in the filter.
When it's time to move him, remove about half of the water from the larger tank and replace it with the old water from the small tank. You will probably want to scoop the water from the top and not transfer any "gunk" that is on the bottom of the tank. Then move as many things as you can from the old tank to the new one: plants, ornaments, filter (or filter media), etc. Do NOT clean these things when you move them. Then move the axolotl into the new tank.