The most readily available billiard ball tray is the "basic black" ball tray found almost anywhere you look online. If you don't have a nearby favorite retailer of fine billiard supply items like I do here in Fort Worth, TX, Ebay can be a good source and proves to be very reasonable.
As for cleaners and polishes and waxes and restorers and on and on, you'll quickly see there are as many "opinions" out there as there are ball sets. Read and explore as many options as you have patience and money, but here are my thoughts on the subject of "ball care"....
What I've found, and the best advice I can offer a fellow ball owner, is to use products specially endorsed (notice I didn't say made or manufactured) by the largest supplier these days of phenolic balls - Aramith. They have two products I can recommend without any reservation:
Aramith Billiard Ball Cleaner
--- the one you will use the vast majority of the time and then only a fingertip dab per ball no matter if you use a dedicated ball polishing machine like the one Diamond offers, or if you prefer to do them by hand with a microfiber towel. Less is better. And buff each one lightly with a clean microfiber towel to ensure they have no residue whatsoever. Play on.
Aramith Billiard Ball Restorer
--- the one you would use IF and only IF you need or find or acquire balls that are in "well used" condition that when you look at them, you'd say to yourself "these need more than cleaning". The restorer is obviously a different product and I'm not qualified to discuss the detailed specs - but it does work beautifully for neglected and abused balls.
IMPORTANT HOWEVER - is to realize that when you use the Restorer, that should be Step ONE. Step TWO would simply be to use the Cleaner product to finely polish / clean the balls you just "restored" to attain that best finish and clean surface.
On a personal note, I would avoid any other products including car waxes, plastic polishes or household cleansers of any kind. Don't even think about dish soap or the Magic Erasures (yes, I have tried them all and conducted my own Dr Dave styled testing using a stimpmeter and observing ball collisions, static electricity and appearance). Nothing works as well as the Aramith products. In my opinion anyway. Again, ebay whatever you need or visit your favorite billiard supply man
One last thought on "equipment": I would avoid any ball polishing set up that has the balls making contact with one another during the polish process. The Diamond polisher had the simplest and best design so far. Expensive? That's relative.