Cut a granite counter top
I have done this with granite tiles in two different sizes, 12 by 12 inches (30x30cm) and 18 by 26 inches (45x65cm).  Not a full slab, but it should basically work the same.

Tools

You have two tasks involved: cutting it, and smoothing/polishing the edges.

For both cutting and polishing, I used an angle grinder.  In my case, I bought a 4-1/2 inch grinder like this one for very cheap from a discount tool import store (about US$20; works great).



For straight cuts, I used a diamond blade (metal blade embedded with diamond chips).  



I clamped a board to the granite to use as a guide rail or fence, and ran the grinder along using a guard like this:



For the round sink opening, I cut out most of it with a series of straight cuts using the diamond blade (inside my desired line), and then smoothed out the curve using a masonry grinding wheel on the angle grinder.

I wore a respirator dust mask and ran the shop vac while cutting, to collect as much dust as possible.

To smooth and polish the edges, I used polishing/"surface conditioning disks" designed for stone and masonry, in the angle grinder.  They come in a set of various grains from coarse to fine.  Just like sanding, start with the coarse and work your way up to the fine.



All of the above parts were done dry; no water involved.

I finished by hand with some diamond polishing pads that come mounted on spongy rubber holders.  These are done wet; using water to lubricate the process and wash away grit while you are doing it.  I used an old plastic milk jug and poked a pin-hole in the side near the bottom, and let it dribble out water continuously while polishing.



All in all, I would say it was easy and inexpensive; it just took time and created a lot of dust.  Get a proper respirator face mask and don't use those little paper masks; your sinuses and lungs are worth proper protection.  



I'm sure I spent less than US$100 on the grinder and cutting/grinding/polishing supplies.  I already had the respirator but they aren't too expensive either.

For the 12 inch tiles, I also used a small tile wet saw, built like a tiny table saw.  It produced nice clean cuts, but since that involves moving the tiles on the saw, it didn't work for the 26 inch tiles, and would not work for slab either.

Results

I used granite tiles for kitchen and bathroom counter tops.  At the time I did the kitchen, I was only aware of square granite tiles in relatively small sizes like the 12 inch ones here.  When I later did the bathroom, I discovered the larger size tiles, which were large enough to run a full tile from back to front edges.  This 5 foot wide counter is only four tiles for the main surface.

Here is the kitchen with 12 inch granite tiles:







And the bathroom with 18 by 26 inch granite tiles: