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Can you "polish" all rocks... or are there only certain rocks which will work? Some of my students want to polish some rocks they found, like the ones their families bought while in the mountains.
The best material to polish is rocks of uniform hardness. Hard (like agate, quartz, etc.) or soft (like marble, etc.) doesn't matter all that much - just don't mix hardnesses in any individual rock. As a general rule of thumb, the "softer" materials (usually calcium-based like marble) are easier to polish, but much less durable - the polish will chip and/or dull with time. Silica-based materials (agate, opal, obsidian. etc.) take longer to polish, but retain the polish for a longer time (but still dull soon or later - thanks to the 4th Law of GeoFantasy). Click here for a related question concerning countertops.
For flat slabs (bookends, coasters, etc.), you have 2 choices: 1) the manual approach or 2) buy yourself a labor-saving device. But vibrating "flat lap" polishers can be spendy (easily $100 or more depending on size and quality). I find that, in an absence of $$$, a piece of plate glass, some silicon carbide grit, and a bit of rock 'n roll for motivation do quite well. Whatever you use, start with a coarse grit (80 or so), a bit of water, and put it on the lap (or rotate the slab on the glass in small circles) until the slab's surface is even. Go to progressively finer grits (down to 600) and then a polishing compound (tin oxide for hard material, and cerium oxide for softer stuff works well).
BE SURE TO COMPLETELY CLEAN OFF ALL THE OLD GRIT WHEN CHANGING TO A FINER GRIT!! Even a small amount can mess up the next stage. This is especially important when changing to the final polishing stage, which should be completed on a felt mat - not the metal plate of the lap, or the glass if you're using the manual polishing method. Most vibrating lap polishers include a felt mat for this purpose.
How do you know when to change to a finer grit? A good trick is to take a soft pencil and draw a cross-hatched grid on the slab. Put it back on the flat lap (or the glass) for a couple minutes. If all the pencil marks are gone, you're ready for the next finer grit. If some of the grid is still visible (commonly in the center), well, at least you now know where the low spots are. Put the slab back on for some more grinding and try the grid test again.
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