If you can, try to move the filter media (mainly the biological media- ceramic rings, bio-balls, or sponges) into the new filter. If you move enough media, then it should allow the new tank to be cycled very quickly. You should be able to add your fish right away if you do this. Of course testing the parameters is the safest way (to make sure there are no problems), but it is probably not necessary.
As for the actual transfer, I would get the new tank filled up and completely set up/running, then catch the fish. If the water in the new tank is different (if it comes from a different water source) then you should probably acclimate it to the new tank (drip acclimation is the best way). If it's the same water, then you should be able to just move the fish right into the new tank. Try to make sure they're both near the same temperatures.