Here is some basic info on tools/abrasives for finishing (hand tools): http://dondougan.homestead.com/ThePr...3_History.html
and power tool abrasives: http://dondougan.homestead.com/ThePr...4_History.html
I don't polish quartz in crystal or agate form often (or on a large scale), but the diamond pads (hand or power) is what I usually use. I have polished obsidian with silicon carbide, but it requires a bit more patience. Cerium oxide (available at places that sell supplies for glass working) is used as the final polishing agent for glass, obsidian, quartz. Travertine (and other carbonate stones such as marble or limestone) would use tin oxide as the final polishing agent.
Whatever the abrasive used, it is simply a matter of applying each abrasive grit to remove the scratches left by the previous grit. It is necessary to use water (or possibly oil in the case of quartz or agate) as a constant rinsing agent during the process when you work with any abrasive finer than about 120 to 150 grit. After each abrasive grit has been used, it is a good idea to rinse the workpiece in clean water, dry the surface, and examine it to make sure all of the previous grit's scratches are removed -- if some coarser grit scratches are missed/skipped-over with the next grit, then the successively finer grits will not remove those scratches (and the scratches will become more obvious as the polish develops).
One way to aid your inspection is to try to use rubbing strokes in alternating directions across the surface for sequential grits - i.e., with 220 grit you might rub in vertical strokes, then with 240 grit you'd rub in horizontal strokes, then with 280 you'd rub vertical strokes again, and then with 320 it would be horizontal, and so forth. Obviously there are places in each carving where it is impractical to alternate the directions of strokes (such as grooved areas), but using the alternating method wherever you can will help you determine how much 'rubbing' with each grit is required to remove the previous grit's scratches.
Power tools will speed up the tedious process to some degree, but at the expense of control. Powertools usually do one specific operation very well, but are usually not flexible enough to get at all the areas that need to be worked -- thus requiring another powertool with a different configuration for each specific area. I use powertools when I can, but there are almost always some areas in each carving where the only viable approach is with hand methods.
You might find some carvers working stones similar to what you are working on this forum as well: http://aboutstone.org/vl/
There is a database of workers in stone, and searchable archives of conversations/threads on many stone-related topics. You can sign-up for free and ask the forum questions and receive answers via email. Other members of the stone list may be able to answer your questions about quartz/agate more fully than I (as I work marble and limestone primarily).