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It is difficult to polish odd shaped pieces of stone by hand. Achieving a polish requires going through several steps: grinding, sanding, pre-polish, and then polish - using "grits" or some sort of abrasive materials in finer and finer particle sizes, such as finer and finer sandpaper. And it takes quite a bit of physical force to do the grinding, etc., - either using grinding wheels or some sort of stone tumbler. Doing it by hand would take a looooong time! :~}
If your pieces of quartz are all about the same size, and are not too large - say no digger than about 2" in diam. - then your best bet would be to get a rock tumbler and tunmble the pieces in that. There are two types of tumblers available these days, the old-fashioned rotating barrel tumblers and the newer vibrating tumblers.
In the rotating tumblers, you load the barrel with your stones and a mixture of tumbling grits and water - a "slurry" - and maybe some buffering material, such as plastic pellets or cut-up rubber bands if the material is delicate or you need to fill out a short load. Then you just turn the tumbler on and let it sit there day in and day out for a few weeks, the grit slurry grinding away at the edges and faces of the stones as they tumble around in the barrel. Once the coarse grit grinding stage has gone as far as it can take the stone, you clean out the tumbler - cleaning up the pieces of stone and the tumbler barrel very carefully, so that you are sure there is no grit - not even a single speck - left in the barrel or on the stones. Then you load the stones back in the tumber with a medium sanding grit sluury and tumble the stones in that. In another few weeks the stones will be as sanded as they are going to get, so it is time to empty out the tumbler again, go through the "surgical cleaning" procedure (which is THE single most important step in tumbling rocks...)- and then put the stones back in with a fine sanding slurry and tumble them in that. When that's done, you go to the pre-polish pahes; then on to the polishing phase. So in about three or four months (maybe more) you have stones that have been ground roughly round, sanded down, and polished.
A vibrating tumbler works in the same way, but quite a bit faster. It usually only takes a week or two of tumbling at each stage to get the stones ground, sanded and polished. The main difference between this and the rotary tumbler (other than speed) is that the rotatary tumbler grinds the stones into roughly rounded or ovate shapes, while a vibrating tumbler just polishes the pieces without changing their shapes very much - the corners get sort of rounded over, but that's it. So if you weant to retain the approximate shape of the stones you are polishing, a rotary tumbler is the way to go. Be advised, though, that vibrating tumblers are quite a bit more expensive than rotary ones.
Another option is to use a set of lapidary grinding wheels and a polishing buff to do basically the same thing, but by hand - where you have some control over what happens. (And which also means you have to develop some skill at the procedure.) Lapidary wheels are not the same thing as your typical workshop's grinding wheels, which are for just grinding metal tool edges. Lapidary wheels are made of different materials - or at least using different "formulas" - than sharpening/cleaning wheels. They are designed to work stone with. And a lapidary wheel set also includes a polishing buff on the end of the shaft - a flat plate to which various types of polishing materials are fixed for the final step of the process.
It is rather difficult to grind, sand, and polish odd shaped pieces of stone using this method. It can be done, after a fashion, but not without some good skill and a bit of luck. Most rockhounds probably wouldn't try to do it this way - opting for the tumbler method, or perhaps the method I'll outline below. But if the pieces of stone lend themselves to being worked on wheels, then it is an option. (Chunks which have concave surfaces on them can't be worked - at least not very well - on a set of lapidary wheels...)
The final way to work odd shaped stones is to use small grinding, sanding, and polishing wheels and buffs on a Dremel-style tool or a close relative to that, the flexible shaft grinder. (Most rockhounds prefer flexible shaft grinders - they are much better to work with, and can do a better job.) A lot of free-form work in stone is done this way. It is, though, a slow and exacting process - even more so than the lapidary wheel set. You have to carefully work one small area of the stone at a time until you have run the entire stone through each step of the process - coarse grinding, sanding, and polishing steps. It requires the most skill of the three different approaches. But it also gives you the most control over what happens to the stone as you work it. (Which is why some lapidary artisans prefer this method over the others.)
As to your other question - about cutting or breaking quartz by hand - no, there really isn't any way to easily break quartz along a straight line without using power tools - namely a diamond-bladed rock saw. Quartz has a conchoidal fracture - which means it breaks in a scalloped fashion, curved - so any use of hammer and chisel will produce at least some curvature when the stone breaks. The only way to prevent this is to actually cut through the stone in some way - and the way most oftened used by rockhounds is the rock saw.
There is a link to a list of rock and mineral clubs on the Table of Contents page here at Bob's Rockshop. You should find that and check out the listings to see if there is a mineral club anywhere near you. If there is, then your best bet for learning how to work stone is to visit the club and get to know the lapidary artisans in it. Most clubs welcome guest, and most lapidary artisans love to share their hobby with new people. So the odds are that you'll find all the help you need at a club.
Plus you are always welcome to ask for advice here. The gang here is like a club of sorts, and we all love to help a new person out if we can.