Most veneers these days are cut to a standard thickness of .5 mm. The old US standard was 1/28", about .7-.8 mm. It is also possible to get veneers cut to .3 mm. It is surprising how accurately they are cut. It's fairly easy to make a strip cutter to slice the pieces you need. I like to cut the strips to 1 mm wide or more, and then thickness a whole layer once they have been glued up, but that's just my preference. Various colors of natural wood are available, as are dyed veneers in a number of colors, and you can also dye your own if you like. In earlier times people made their own veneers. Some violin makers still make the veneers for purfling strips.
To do this you start with a piece of clear, white wood: Italian poplar is traditional. Prepare a thin plank of a usable length; for violin purfling, say, 1/4" thick by 10" long or so. Set up a sharp plane to take a heavy cut, and use it to 'shoot' strips off the side of the plank. It helps to slightly dampen the wood before you take the cut, which prevents the chip breaker from breaking the wood fibers as the shaving is peeled off. Adjust the plane to take a shaving the desired thickness: for violin purfling that would be about .6 mm for the white line and .3 mm for the black. The thin shavings for the black lines are dyed by boiling them in a solution of ferric acetate (vinegar and nails) and tannin. Strad boiled his in a copper trough: his purfling turns a little green.
Eugene Clarke and Jonathon Peterson published a really useful pair of articles in American Lutherie #71 and #73, Fall of '02 and Spring of '02, which should be out in the appropriate 'Big Red Book' from the GAL.