Unlike Frank I have polished everything with diamond. My quartz competition stones were polished with diamond. The old guy that taught me to use lava soap also said diamond will polish everything. Might not be entirely true but close. I started out using diamond on a tin lap. Batt lap is a harder tin lap and is better. For quartz I cut on 600 topper then 3,000 on batt and finish on a 6″ Matrix with cerium oxide. I have found nothing quicker or better than the Matrix and a small amount of cerium for quartz. Get one other Batt to use with 50,000 diamond for everything else. Two Batts and the Matrix and you will need nothing else for polishing starting out, at least that is what I have found works for me. I have no use for any cutting lap finer than 600. The cheap Chinese toppers work fine for me on a master lap. Get 2 more toppers in 260 and 100 for roughing out the stone. The 100 is only used sparingly and be sure and leave enough for the 260 to remove the 100 grit scratches. Before polishing be sure and remove all scratches with the Batt and 3,000 and your meets will not move at final polish, or at least very little until you learn more. I used those flimsy plastic ultra laps when first starting out and they will work but are much too slow for me. It does not take but a few seconds on a Matrix with cerium in water with an artist brush with a small drip of water after pre-polishing on a 3,000 Batt to polish a quartz stone. Some like those bonded oxide laps for polishing. i have tried them and like the ultra-laps they will work but more slowly than Matrix with cerium. One advantage in using them is less mess but it takes so little cerium on a Matrix to polish there is very little mess.
While learning always remember to have fun and enjoy. You will have some trial and errors but with practice you will be competing one day. Get some cheap man made quartz and do not fret too much when it does not turn out like you want. Take it off the dop and start another one.