If you transfer your existing live rock, your tank will now be instantly cycled enough to handle that tank's bioload, so you're good there. Don't transfer any existing sand.
Start by taking some water from the old tank (before you stir it up) and putting it in a bucket with a heater, some form of circulation, A LID, and ideally a small rock or other in-tank object to hide behind. Ideally don't fill the bucket all the way, to help prevent jumping out. As you catch mobile livestock, put it in there. Also, make sure the new tank isn't so full that adding rocks (and your arms) will overflow it.
Catch whatever animals you easily can, then begin to move the rockwork into the new tank, piece by piece, catching livestock whenever you get the chance. Preferably catch all the livestock out before you move the bottom rocks, in case there's anything nasty under there. Either set the lid on top of the bucket whenever it's not actively being added to, or be vigilant in monitoring that all the fish in the bucket stay in the bucket. Or both.
Once all the rockwork is settled in the new tank, you should, assuming the tanks are at the same salinity and you've gotten the new tank up to normal temperature, be able to simply pour the fish in. Double-check before doing so that all fish are in the bucket and not on the floor.
