The first cleaning of any new gun should include a thorough degreasing with gun scrubber or a plastic safe brake parts cleaner.
For the Judge, you will want to remove the grip and the side plate. Use copious amounts of gun scrubber to remove any trace of packing grease or oil (seems they use an oil these days) since it can gum up the works. Replace the side plate and grip and cock the hammer. Put 2 or 3 drops of rem-oil, or any other light gun oil that migrates well, on the hammer. Dry fire the gun 10 to 20 times. Place one drop of the oil under the extractor star, and one drop on the ejector rod where it protrudes from the cylinder. That's just about all the lube you will need.
Add to all that a normal cleaning, and you are all set.