Hi, Matt - I have a Krome, given to me by a very good friend. Using it is a bit tricky, it's capable of doing the finest of detail work, I'm afraid my skills are not equal to the abilities of the AB, but it does work extremely well and I enjoy it.
First thing, go check on "Don's Airbrush" website, all things airbrush are found there, including your Krome review and cleaning information. Best time investment you'll make, when getting familiar with your AB
MM66 is spot on, clean it well after each use. Those tiny precision parts don't take kindly to dried paint or other gunk setting up in it. I spray two rinse cups through it, take the front bits off and soak them in lacquer thinner, then wipe down the forward body interior and cup with q-tips, paper towel bits and thinner. I RARELY remove the tiny tip, and it doesn't seem to make any difference whether I do or not.
For cleaning the forward body interior I use interdental cleaning devices, they are covered on Don's site as well. For cleaning the nozzle and tip I use small brushes and a toothpick sanded down to the diameter of the tip small end. That will gently remove any paint resdue inside, but not harm the tip itself. Very little pressure is required, the end of the tip is delicate and should never have anything pressed hard in it.
Like 66 I use Regdab for the needle, a wipe down of it and then a bit of the lube on the forward part of the needle will help the needle and needle bearing. Remember, the needle bearing is adjustable with a tiny screwdriver, but it is sensitive and you may have to try a few times to get it right.
Best of luck with your Krome, it is an excellent bit of precision.